TPTP Problem File: SLH0815^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Prefix_Free_Code_Combinators/0000_Prefix_Free_Code_Combinators/prob_00542_018387__11990496_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1414 ( 826 unt; 133 typ; 0 def)
% Number of atoms : 3276 (1909 equ; 0 cnn)
% Maximal formula atoms : 81 ( 2 avg)
% Number of connectives : 9242 ( 378 ~; 104 |; 174 &;7601 @)
% ( 0 <=>; 985 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 5 avg)
% Number of types : 21 ( 20 usr)
% Number of type conns : 383 ( 383 >; 0 *; 0 +; 0 <<)
% Number of symbols : 116 ( 113 usr; 19 con; 0-8 aty)
% Number of variables : 2817 ( 73 ^;2692 !; 52 ?;2817 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 10:02:48.025
%------------------------------------------------------------------------------
% Could-be-implicit typings (20)
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Extended____Real__Oereal_Mt__Extended____Real__Oereal_J_J,type,
list_P6029375857206349485_ereal: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Extended____Real__Oereal_J_J,type,
list_P888892035181795709_ereal: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Extended____Real__Oereal_Mtf__a_J_J,type,
list_P546110976589770589real_a: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
list_P1396940483166286381od_a_a: $tType ).
thf(ty_n_t__List__Olist_It__Extended____Real__Oereal_J,type,
list_Extended_ereal: $tType ).
thf(ty_n_t__Set__Oset_It__Extended____Real__Oereal_J,type,
set_Extended_ereal: $tType ).
thf(ty_n_t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
option_list_o: $tType ).
thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
list_real: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Extended____Real__Oereal,type,
extended_ereal: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Typerep__Otyperep,type,
typerep: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (113)
thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity_001t__Extended____Real__Oereal,type,
extend1530274965995635425_ereal: extended_ereal ).
thf(sy_c_Extended__Real_Oereal_OMInfty,type,
extended_MInfty: extended_ereal ).
thf(sy_c_Extended__Real_Oereal_OPInfty,type,
extended_PInfty: extended_ereal ).
thf(sy_c_Extended__Real_Oereal_Oereal,type,
extended_ereal2: real > extended_ereal ).
thf(sy_c_Extended__Real_Oereal_Osize__ereal,type,
extended_size_ereal: extended_ereal > nat ).
thf(sy_c_Extended__Real_Oreal__of__ereal,type,
extend2982805604970551563_ereal: extended_ereal > real ).
thf(sy_c_Fun_Ocomp_001t__Extended____Real__Oereal_001_Eo_001t__Extended____Real__Oereal,type,
comp_E6928223901766943329_ereal: ( extended_ereal > $o ) > ( extended_ereal > extended_ereal ) > extended_ereal > $o ).
thf(sy_c_Fun_Ocomp_001t__Extended____Real__Oereal_001_Eo_001tf__a,type,
comp_E512332703494744593al_o_a: ( extended_ereal > $o ) > ( a > extended_ereal ) > a > $o ).
thf(sy_c_Fun_Ocomp_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal_001tf__a,type,
comp_E1870838029643375451real_a: ( extended_ereal > extended_ereal ) > ( a > extended_ereal ) > a > extended_ereal ).
thf(sy_c_Fun_Ocomp_001t__Extended____Real__Oereal_001tf__a_001tf__a,type,
comp_E4980989167044428331al_a_a: ( extended_ereal > a ) > ( a > extended_ereal ) > a > a ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_It__Extended____Real__Oereal_J_001t__List__Olist_It__Extended____Real__Oereal_J_001t__List__Olist_Itf__a_J,type,
comp_l1138752672294181729list_a: ( list_Extended_ereal > list_Extended_ereal ) > ( list_a > list_Extended_ereal ) > list_a > list_Extended_ereal ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_It__Extended____Real__Oereal_J_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
comp_l865004379364615857list_a: ( list_Extended_ereal > list_a ) > ( list_a > list_Extended_ereal ) > list_a > list_a ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_Itf__a_J_001t__List__Olist_It__Extended____Real__Oereal_J_001t__List__Olist_Itf__a_J,type,
comp_l2696117608003671697list_a: ( list_a > list_Extended_ereal ) > ( list_a > list_a ) > list_a > list_Extended_ereal ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
comp_l467081738950601185list_a: ( list_a > list_a ) > ( list_a > list_a ) > list_a > list_a ).
thf(sy_c_Fun_Ocomp_001tf__a_001_Eo_001t__Extended____Real__Oereal,type,
comp_a5376481364045625617_ereal: ( a > $o ) > ( extended_ereal > a ) > extended_ereal > $o ).
thf(sy_c_Fun_Ocomp_001tf__a_001_Eo_001tf__a,type,
comp_a_o_a: ( a > $o ) > ( a > a ) > a > $o ).
thf(sy_c_Fun_Ocomp_001tf__a_001t__Extended____Real__Oereal_001tf__a,type,
comp_a6852332661818589707real_a: ( a > extended_ereal ) > ( a > a ) > a > extended_ereal ).
thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001tf__a,type,
comp_a_a_a: ( a > a ) > ( a > a ) > a > a ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Extended____Real__Oereal,type,
abs_ab7465543570706387889_ereal: extended_ereal > extended_ereal ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
abs_abs_real: real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Real__Oereal,type,
minus_2816186181549245109_ereal: extended_ereal > extended_ereal > extended_ereal ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Real__Oereal,type,
one_on4623092294121504201_ereal: extended_ereal ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Real__Oereal,type,
plus_p7876563987511257093_ereal: extended_ereal > extended_ereal > extended_ereal ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Real__Oereal,type,
times_7703590493115627913_ereal: extended_ereal > extended_ereal > extended_ereal ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Extended____Real__Oereal,type,
uminus27091377158695749_ereal: extended_ereal > extended_ereal ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Real__Oereal,type,
zero_z2744965634713055877_ereal: extended_ereal ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Extended____Real__Oereal,type,
groups3567983573054521703_ereal: list_Extended_ereal > extended_ereal ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Int__Oint,type,
groups4559388385066561235st_int: list_int > int ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
groups4561878855575611511st_nat: list_nat > nat ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Real__Oreal,type,
groups6723090944982001619t_real: list_real > real ).
thf(sy_c_Groups__List_Omonoid__list_OF_001t__Extended____Real__Oereal,type,
groups4041799636089507985_ereal: ( extended_ereal > extended_ereal > extended_ereal ) > extended_ereal > list_Extended_ereal > extended_ereal ).
thf(sy_c_Groups__List_Omonoid__list_OF_001t__Int__Oint,type,
groups_monoid_F_int: ( int > int > int ) > int > list_int > int ).
thf(sy_c_Groups__List_Omonoid__list_OF_001t__Nat__Onat,type,
groups_monoid_F_nat: ( nat > nat > nat ) > nat > list_nat > nat ).
thf(sy_c_Groups__List_Omonoid__list_OF_001t__Real__Oreal,type,
groups_monoid_F_real: ( real > real > real ) > real > list_real > real ).
thf(sy_c_If_001t__Extended____Real__Oereal,type,
if_Extended_ereal: $o > extended_ereal > extended_ereal > extended_ereal ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_List_Ofilter_001t__Extended____Real__Oereal,type,
filter116404481290734027_ereal: ( extended_ereal > $o ) > list_Extended_ereal > list_Extended_ereal ).
thf(sy_c_List_Ofilter_001tf__a,type,
filter_a: ( a > $o ) > list_a > list_a ).
thf(sy_c_List_Ofoldr_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal,type,
foldr_6806735636647425191_ereal: ( extended_ereal > extended_ereal > extended_ereal ) > list_Extended_ereal > extended_ereal > extended_ereal ).
thf(sy_c_List_Ofoldr_001t__Int__Oint_001t__Int__Oint,type,
foldr_int_int: ( int > int > int ) > list_int > int > int ).
thf(sy_c_List_Ofoldr_001t__Nat__Onat_001t__Nat__Onat,type,
foldr_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).
thf(sy_c_List_Ofoldr_001t__Real__Oreal_001t__Real__Oreal,type,
foldr_real_real: ( real > real > real ) > list_real > real > real ).
thf(sy_c_List_Ogen__length_001t__Extended____Real__Oereal,type,
gen_le964406056475918689_ereal: nat > list_Extended_ereal > nat ).
thf(sy_c_List_Ogen__length_001tf__a,type,
gen_length_a: nat > list_a > nat ).
thf(sy_c_List_Olist_ONil_001t__Extended____Real__Oereal,type,
nil_Extended_ereal: list_Extended_ereal ).
thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
nil_int: list_int ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Real__Oreal,type,
nil_real: list_real ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Omap_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal,type,
map_Ex83193753010212676_ereal: ( extended_ereal > extended_ereal ) > list_Extended_ereal > list_Extended_ereal ).
thf(sy_c_List_Olist_Omap_001t__Extended____Real__Oereal_001tf__a,type,
map_Extended_ereal_a: ( extended_ereal > a ) > list_Extended_ereal > list_a ).
thf(sy_c_List_Olist_Omap_001tf__a_001t__Extended____Real__Oereal,type,
map_a_Extended_ereal: ( a > extended_ereal ) > list_a > list_Extended_ereal ).
thf(sy_c_List_Olist_Omap_001tf__a_001t__Real__Oreal,type,
map_a_real: ( a > real ) > list_a > list_real ).
thf(sy_c_List_Olist_Omap_001tf__a_001tf__a,type,
map_a_a: ( a > a ) > list_a > list_a ).
thf(sy_c_List_Olist_Oset_001t__Extended____Real__Oereal,type,
set_Extended_ereal2: list_Extended_ereal > set_Extended_ereal ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Omap__tailrec_001tf__a_001t__Extended____Real__Oereal,type,
map_ta5354146704884569822_ereal: ( a > extended_ereal ) > list_a > list_Extended_ereal ).
thf(sy_c_List_Omap__tailrec_001tf__a_001tf__a,type,
map_tailrec_a_a: ( a > a ) > list_a > list_a ).
thf(sy_c_List_Oproduct_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal,type,
produc2704682175734110533_ereal: list_Extended_ereal > list_Extended_ereal > list_P6029375857206349485_ereal ).
thf(sy_c_List_Oproduct_001t__Extended____Real__Oereal_001tf__a,type,
produc6916318822199699189real_a: list_Extended_ereal > list_a > list_P546110976589770589real_a ).
thf(sy_c_List_Oproduct_001tf__a_001t__Extended____Real__Oereal,type,
produc2373813038293282581_ereal: list_a > list_Extended_ereal > list_P888892035181795709_ereal ).
thf(sy_c_List_Oproduct_001tf__a_001tf__a,type,
product_a_a: list_a > list_a > list_P1396940483166286381od_a_a ).
thf(sy_c_List_Oremove1_001tf__a,type,
remove1_a: a > list_a > list_a ).
thf(sy_c_List_Osplice_001t__Extended____Real__Oereal,type,
splice291902714325861031_ereal: list_Extended_ereal > list_Extended_ereal > list_Extended_ereal ).
thf(sy_c_List_Osplice_001tf__a,type,
splice_a: list_a > list_a > list_a ).
thf(sy_c_Nat_Osize__class_Osize_001t__Extended____Real__Oereal,type,
size_s5012395470859603514_ereal: extended_ereal > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Extended____Real__Oereal_J,type,
size_s2768339837476157504_ereal: list_Extended_ereal > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Extended____Real__Oereal_Mt__Extended____Real__Oereal_J_J,type,
size_s2383823777779303705_ereal: list_P6029375857206349485_ereal > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Extended____Real__Oereal_Mtf__a_J_J,type,
size_s6587793725349607369real_a: list_P546110976589770589real_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Extended____Real__Oereal_J_J,type,
size_s6930574783941632489_ereal: list_P888892035181795709_ereal > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
size_s3885678630836030617od_a_a: list_P1396940483166286381od_a_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Typerep__Otyperep,type,
size_size_typerep: typerep > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Real__Oereal,type,
ord_le1188267648640031866_ereal: extended_ereal > extended_ereal > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Real__Oereal,type,
ord_le1083603963089353582_ereal: extended_ereal > extended_ereal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Prefix__Free__Code__Combinators_OLf_092_060_094sub_062e_001tf__a,type,
prefix4097710381326367690Lf_e_a: ( a > option_list_o ) > nat > list_a > option_list_o ).
thf(sy_c_Prefix__Free__Code__Combinators_Obit__count,type,
prefix3213528784805800034_count: option_list_o > extended_ereal ).
thf(sy_c_Prefix__Free__Code__Combinators_Oopt__append,type,
prefix5314359684614007693append: option_list_o > option_list_o > option_list_o ).
thf(sy_c_String_Ochar_OChar,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_member_001t__Extended____Real__Oereal,type,
member2350847679896131959_ereal: extended_ereal > set_Extended_ereal > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_e,type,
e: a > option_list_o ).
thf(sy_v_xsa____,type,
xsa: list_a ).
% Relevant facts (1271)
thf(fact_0_sum__list__0,axiom,
! [Xs: list_a] :
( ( groups3567983573054521703_ereal
@ ( map_a_Extended_ereal
@ ^ [X: a] : zero_z2744965634713055877_ereal
@ Xs ) )
= zero_z2744965634713055877_ereal ) ).
% sum_list_0
thf(fact_1_length__map,axiom,
! [F: extended_ereal > a,Xs: list_Extended_ereal] :
( ( size_size_list_a @ ( map_Extended_ereal_a @ F @ Xs ) )
= ( size_s2768339837476157504_ereal @ Xs ) ) ).
% length_map
thf(fact_2_length__map,axiom,
! [F: extended_ereal > extended_ereal,Xs: list_Extended_ereal] :
( ( size_s2768339837476157504_ereal @ ( map_Ex83193753010212676_ereal @ F @ Xs ) )
= ( size_s2768339837476157504_ereal @ Xs ) ) ).
% length_map
thf(fact_3_length__map,axiom,
! [F: a > extended_ereal,Xs: list_a] :
( ( size_s2768339837476157504_ereal @ ( map_a_Extended_ereal @ F @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_map
thf(fact_4_length__map,axiom,
! [F: a > a,Xs: list_a] :
( ( size_size_list_a @ ( map_a_a @ F @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_map
thf(fact_5_map__ident,axiom,
( ( map_a_a
@ ^ [X: a] : X )
= ( ^ [Xs2: list_a] : Xs2 ) ) ).
% map_ident
thf(fact_6_map__eq__imp__length__eq,axiom,
! [F: a > a,Xs: list_a,G: a > a,Ys: list_a] :
( ( ( map_a_a @ F @ Xs )
= ( map_a_a @ G @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_7_map__eq__imp__length__eq,axiom,
! [F: a > extended_ereal,Xs: list_a,G: extended_ereal > extended_ereal,Ys: list_Extended_ereal] :
( ( ( map_a_Extended_ereal @ F @ Xs )
= ( map_Ex83193753010212676_ereal @ G @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_s2768339837476157504_ereal @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_8_map__eq__imp__length__eq,axiom,
! [F: a > a,Xs: list_a,G: extended_ereal > a,Ys: list_Extended_ereal] :
( ( ( map_a_a @ F @ Xs )
= ( map_Extended_ereal_a @ G @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_s2768339837476157504_ereal @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_9_map__eq__imp__length__eq,axiom,
! [F: extended_ereal > extended_ereal,Xs: list_Extended_ereal,G: a > extended_ereal,Ys: list_a] :
( ( ( map_Ex83193753010212676_ereal @ F @ Xs )
= ( map_a_Extended_ereal @ G @ Ys ) )
=> ( ( size_s2768339837476157504_ereal @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_10_map__eq__imp__length__eq,axiom,
! [F: extended_ereal > a,Xs: list_Extended_ereal,G: a > a,Ys: list_a] :
( ( ( map_Extended_ereal_a @ F @ Xs )
= ( map_a_a @ G @ Ys ) )
=> ( ( size_s2768339837476157504_ereal @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_11_map__eq__imp__length__eq,axiom,
! [F: a > extended_ereal,Xs: list_a,G: a > extended_ereal,Ys: list_a] :
( ( ( map_a_Extended_ereal @ F @ Xs )
= ( map_a_Extended_ereal @ G @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_12_list_Omap__ident,axiom,
! [T: list_a] :
( ( map_a_a
@ ^ [X: a] : X
@ T )
= T ) ).
% list.map_ident
thf(fact_13_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_Extended_ereal] :
( ( size_s2768339837476157504_ereal @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_14_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_a] :
( ( size_size_list_a @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_15_neq__if__length__neq,axiom,
! [Xs: list_Extended_ereal,Ys: list_Extended_ereal] :
( ( ( size_s2768339837476157504_ereal @ Xs )
!= ( size_s2768339837476157504_ereal @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_16_neq__if__length__neq,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
!= ( size_size_list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_17_size__neq__size__imp__neq,axiom,
! [X2: list_Extended_ereal,Y: list_Extended_ereal] :
( ( ( size_s2768339837476157504_ereal @ X2 )
!= ( size_s2768339837476157504_ereal @ Y ) )
=> ( X2 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_18_size__neq__size__imp__neq,axiom,
! [X2: list_a,Y: list_a] :
( ( ( size_size_list_a @ X2 )
!= ( size_size_list_a @ Y ) )
=> ( X2 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_19_size__neq__size__imp__neq,axiom,
! [X2: char,Y: char] :
( ( ( size_size_char @ X2 )
!= ( size_size_char @ Y ) )
=> ( X2 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_20_size__neq__size__imp__neq,axiom,
! [X2: typerep,Y: typerep] :
( ( ( size_size_typerep @ X2 )
!= ( size_size_typerep @ Y ) )
=> ( X2 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_21_size__neq__size__imp__neq,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( ( size_s5012395470859603514_ereal @ X2 )
!= ( size_s5012395470859603514_ereal @ Y ) )
=> ( X2 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_22_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_23_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_24_zero__reorient,axiom,
! [X2: extended_ereal] :
( ( zero_z2744965634713055877_ereal = X2 )
= ( X2 = zero_z2744965634713055877_ereal ) ) ).
% zero_reorient
thf(fact_25_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_26_zero__reorient,axiom,
! [X2: int] :
( ( zero_zero_int = X2 )
= ( X2 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_27_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_28_sum__list__ereal,axiom,
! [F: a > real,Xs: list_a] :
( ( groups3567983573054521703_ereal
@ ( map_a_Extended_ereal
@ ^ [X: a] : ( extended_ereal2 @ ( F @ X ) )
@ Xs ) )
= ( extended_ereal2 @ ( groups6723090944982001619t_real @ ( map_a_real @ F @ Xs ) ) ) ) ).
% sum_list_ereal
thf(fact_29_length__code,axiom,
( size_size_list_a
= ( gen_length_a @ zero_zero_nat ) ) ).
% length_code
thf(fact_30_length__code,axiom,
( size_s2768339837476157504_ereal
= ( gen_le964406056475918689_ereal @ zero_zero_nat ) ) ).
% length_code
thf(fact_31_map__eq__map__tailrec,axiom,
map_a_Extended_ereal = map_ta5354146704884569822_ereal ).
% map_eq_map_tailrec
thf(fact_32_map__eq__map__tailrec,axiom,
map_a_a = map_tailrec_a_a ).
% map_eq_map_tailrec
thf(fact_33_Infty__neq__0_I1_J,axiom,
extend1530274965995635425_ereal != zero_z2744965634713055877_ereal ).
% Infty_neq_0(1)
thf(fact_34_typerep_Osize__neq,axiom,
! [X2: typerep] :
( ( size_size_typerep @ X2 )
!= zero_zero_nat ) ).
% typerep.size_neq
thf(fact_35_sum__list__map__filter_H,axiom,
! [F: a > extended_ereal,P: a > $o,Xs: list_a] :
( ( groups3567983573054521703_ereal @ ( map_a_Extended_ereal @ F @ ( filter_a @ P @ Xs ) ) )
= ( groups3567983573054521703_ereal
@ ( map_a_Extended_ereal
@ ^ [X: a] : ( if_Extended_ereal @ ( P @ X ) @ ( F @ X ) @ zero_z2744965634713055877_ereal )
@ Xs ) ) ) ).
% sum_list_map_filter'
thf(fact_36_ereal__cong,axiom,
! [X2: real,Y: real] :
( ( X2 = Y )
=> ( ( extended_ereal2 @ X2 )
= ( extended_ereal2 @ Y ) ) ) ).
% ereal_cong
thf(fact_37_ereal_Oinject,axiom,
! [X1: real,Y1: real] :
( ( ( extended_ereal2 @ X1 )
= ( extended_ereal2 @ Y1 ) )
= ( X1 = Y1 ) ) ).
% ereal.inject
thf(fact_38_ereal__eq__0_I2_J,axiom,
! [R: real] :
( ( zero_z2744965634713055877_ereal
= ( extended_ereal2 @ R ) )
= ( R = zero_zero_real ) ) ).
% ereal_eq_0(2)
thf(fact_39_ereal__eq__0_I1_J,axiom,
! [R: real] :
( ( ( extended_ereal2 @ R )
= zero_z2744965634713055877_ereal )
= ( R = zero_zero_real ) ) ).
% ereal_eq_0(1)
thf(fact_40_zero__ereal__def,axiom,
( zero_z2744965634713055877_ereal
= ( extended_ereal2 @ zero_zero_real ) ) ).
% zero_ereal_def
thf(fact_41_ereal_Osize_I4_J,axiom,
! [X1: real] :
( ( size_s5012395470859603514_ereal @ ( extended_ereal2 @ X1 ) )
= zero_zero_nat ) ).
% ereal.size(4)
thf(fact_42_PInfty__neq__ereal_I1_J,axiom,
! [R: real] :
( ( extended_ereal2 @ R )
!= extend1530274965995635425_ereal ) ).
% PInfty_neq_ereal(1)
thf(fact_43_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_44_ereal_Osize__gen_I1_J,axiom,
! [X1: real] :
( ( extended_size_ereal @ ( extended_ereal2 @ X1 ) )
= zero_zero_nat ) ).
% ereal.size_gen(1)
thf(fact_45_sum__list__map__filter,axiom,
! [Xs: list_a,P: a > $o,F: a > extended_ereal] :
( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ~ ( P @ X3 )
=> ( ( F @ X3 )
= zero_z2744965634713055877_ereal ) ) )
=> ( ( groups3567983573054521703_ereal @ ( map_a_Extended_ereal @ F @ ( filter_a @ P @ Xs ) ) )
= ( groups3567983573054521703_ereal @ ( map_a_Extended_ereal @ F @ Xs ) ) ) ) ).
% sum_list_map_filter
thf(fact_46_char_Osize_I2_J,axiom,
! [X1: $o,X22: $o,X32: $o,X4: $o,X5: $o,X6: $o,X7: $o,X8: $o] :
( ( size_size_char @ ( char2 @ X1 @ X22 @ X32 @ X4 @ X5 @ X6 @ X7 @ X8 ) )
= zero_zero_nat ) ).
% char.size(2)
thf(fact_47_length__filter__map,axiom,
! [P: a > $o,F: a > a,Xs: list_a] :
( ( size_size_list_a @ ( filter_a @ P @ ( map_a_a @ F @ Xs ) ) )
= ( size_size_list_a @ ( filter_a @ ( comp_a_o_a @ P @ F ) @ Xs ) ) ) ).
% length_filter_map
thf(fact_48_length__filter__map,axiom,
! [P: a > $o,F: extended_ereal > a,Xs: list_Extended_ereal] :
( ( size_size_list_a @ ( filter_a @ P @ ( map_Extended_ereal_a @ F @ Xs ) ) )
= ( size_s2768339837476157504_ereal @ ( filter116404481290734027_ereal @ ( comp_a5376481364045625617_ereal @ P @ F ) @ Xs ) ) ) ).
% length_filter_map
thf(fact_49_length__filter__map,axiom,
! [P: extended_ereal > $o,F: a > extended_ereal,Xs: list_a] :
( ( size_s2768339837476157504_ereal @ ( filter116404481290734027_ereal @ P @ ( map_a_Extended_ereal @ F @ Xs ) ) )
= ( size_size_list_a @ ( filter_a @ ( comp_E512332703494744593al_o_a @ P @ F ) @ Xs ) ) ) ).
% length_filter_map
thf(fact_50_length__filter__map,axiom,
! [P: extended_ereal > $o,F: extended_ereal > extended_ereal,Xs: list_Extended_ereal] :
( ( size_s2768339837476157504_ereal @ ( filter116404481290734027_ereal @ P @ ( map_Ex83193753010212676_ereal @ F @ Xs ) ) )
= ( size_s2768339837476157504_ereal @ ( filter116404481290734027_ereal @ ( comp_E6928223901766943329_ereal @ P @ F ) @ Xs ) ) ) ).
% length_filter_map
thf(fact_51_sum__list_ONil,axiom,
( ( groups4561878855575611511st_nat @ nil_nat )
= zero_zero_nat ) ).
% sum_list.Nil
thf(fact_52_sum__list_ONil,axiom,
( ( groups6723090944982001619t_real @ nil_real )
= zero_zero_real ) ).
% sum_list.Nil
thf(fact_53_sum__list_ONil,axiom,
( ( groups4559388385066561235st_int @ nil_int )
= zero_zero_int ) ).
% sum_list.Nil
thf(fact_54_sum__list_ONil,axiom,
( ( groups3567983573054521703_ereal @ nil_Extended_ereal )
= zero_z2744965634713055877_ereal ) ).
% sum_list.Nil
thf(fact_55_gen__length__def,axiom,
( gen_length_a
= ( ^ [N2: nat,Xs2: list_a] : ( plus_plus_nat @ N2 @ ( size_size_list_a @ Xs2 ) ) ) ) ).
% gen_length_def
thf(fact_56_gen__length__def,axiom,
( gen_le964406056475918689_ereal
= ( ^ [N2: nat,Xs2: list_Extended_ereal] : ( plus_plus_nat @ N2 @ ( size_s2768339837476157504_ereal @ Xs2 ) ) ) ) ).
% gen_length_def
thf(fact_57_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_58_length__0__conv,axiom,
! [Xs: list_Extended_ereal] :
( ( ( size_s2768339837476157504_ereal @ Xs )
= zero_zero_nat )
= ( Xs = nil_Extended_ereal ) ) ).
% length_0_conv
thf(fact_59_add__right__cancel,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_60_add__right__cancel,axiom,
! [B: real,A: real,C2: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_61_add__right__cancel,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_62_add__left__cancel,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_63_add__left__cancel,axiom,
! [A: real,B: real,C2: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_64_add__left__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_65_char_Oinject,axiom,
! [X1: $o,X22: $o,X32: $o,X4: $o,X5: $o,X6: $o,X7: $o,X8: $o,Y1: $o,Y2: $o,Y3: $o,Y4: $o,Y5: $o,Y6: $o,Y7: $o,Y8: $o] :
( ( ( char2 @ X1 @ X22 @ X32 @ X4 @ X5 @ X6 @ X7 @ X8 )
= ( char2 @ Y1 @ Y2 @ Y3 @ Y4 @ Y5 @ Y6 @ Y7 @ Y8 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 )
& ( X32 = Y3 )
& ( X4 = Y4 )
& ( X5 = Y5 )
& ( X6 = Y6 )
& ( X7 = Y7 )
& ( X8 = Y8 ) ) ) ).
% char.inject
thf(fact_66_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_67_add__0,axiom,
! [A: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ zero_z2744965634713055877_ereal @ A )
= A ) ).
% add_0
thf(fact_68_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_69_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_70_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y ) )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_71_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_72_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_73_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_74_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_75_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_76_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_77_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_78_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_79_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_80_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_81_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_82_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_83_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_84_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_85_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_86_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_87_add_Oright__neutral,axiom,
! [A: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ A @ zero_z2744965634713055877_ereal )
= A ) ).
% add.right_neutral
thf(fact_88_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_89_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_90_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_91_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_92_map__is__Nil__conv,axiom,
! [F: a > extended_ereal,Xs: list_a] :
( ( ( map_a_Extended_ereal @ F @ Xs )
= nil_Extended_ereal )
= ( Xs = nil_a ) ) ).
% map_is_Nil_conv
thf(fact_93_map__is__Nil__conv,axiom,
! [F: a > a,Xs: list_a] :
( ( ( map_a_a @ F @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% map_is_Nil_conv
thf(fact_94_Nil__is__map__conv,axiom,
! [F: a > extended_ereal,Xs: list_a] :
( ( nil_Extended_ereal
= ( map_a_Extended_ereal @ F @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_map_conv
thf(fact_95_Nil__is__map__conv,axiom,
! [F: a > a,Xs: list_a] :
( ( nil_a
= ( map_a_a @ F @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_map_conv
thf(fact_96_list_Omap__disc__iff,axiom,
! [F: a > extended_ereal,A: list_a] :
( ( ( map_a_Extended_ereal @ F @ A )
= nil_Extended_ereal )
= ( A = nil_a ) ) ).
% list.map_disc_iff
thf(fact_97_list_Omap__disc__iff,axiom,
! [F: a > a,A: list_a] :
( ( ( map_a_a @ F @ A )
= nil_a )
= ( A = nil_a ) ) ).
% list.map_disc_iff
thf(fact_98_List_Omap_Ocomp,axiom,
! [F: extended_ereal > extended_ereal,G: a > extended_ereal] :
( ( comp_l1138752672294181729list_a @ ( map_Ex83193753010212676_ereal @ F ) @ ( map_a_Extended_ereal @ G ) )
= ( map_a_Extended_ereal @ ( comp_E1870838029643375451real_a @ F @ G ) ) ) ).
% List.map.comp
thf(fact_99_List_Omap_Ocomp,axiom,
! [F: extended_ereal > a,G: a > extended_ereal] :
( ( comp_l865004379364615857list_a @ ( map_Extended_ereal_a @ F ) @ ( map_a_Extended_ereal @ G ) )
= ( map_a_a @ ( comp_E4980989167044428331al_a_a @ F @ G ) ) ) ).
% List.map.comp
thf(fact_100_List_Omap_Ocomp,axiom,
! [F: a > extended_ereal,G: a > a] :
( ( comp_l2696117608003671697list_a @ ( map_a_Extended_ereal @ F ) @ ( map_a_a @ G ) )
= ( map_a_Extended_ereal @ ( comp_a6852332661818589707real_a @ F @ G ) ) ) ).
% List.map.comp
thf(fact_101_List_Omap_Ocomp,axiom,
! [F: a > a,G: a > a] :
( ( comp_l467081738950601185list_a @ ( map_a_a @ F ) @ ( map_a_a @ G ) )
= ( map_a_a @ ( comp_a_a_a @ F @ G ) ) ) ).
% List.map.comp
thf(fact_102_map__comp__map,axiom,
! [F: extended_ereal > extended_ereal,G: a > extended_ereal] :
( ( comp_l1138752672294181729list_a @ ( map_Ex83193753010212676_ereal @ F ) @ ( map_a_Extended_ereal @ G ) )
= ( map_a_Extended_ereal @ ( comp_E1870838029643375451real_a @ F @ G ) ) ) ).
% map_comp_map
thf(fact_103_map__comp__map,axiom,
! [F: extended_ereal > a,G: a > extended_ereal] :
( ( comp_l865004379364615857list_a @ ( map_Extended_ereal_a @ F ) @ ( map_a_Extended_ereal @ G ) )
= ( map_a_a @ ( comp_E4980989167044428331al_a_a @ F @ G ) ) ) ).
% map_comp_map
thf(fact_104_map__comp__map,axiom,
! [F: a > extended_ereal,G: a > a] :
( ( comp_l2696117608003671697list_a @ ( map_a_Extended_ereal @ F ) @ ( map_a_a @ G ) )
= ( map_a_Extended_ereal @ ( comp_a6852332661818589707real_a @ F @ G ) ) ) ).
% map_comp_map
thf(fact_105_map__comp__map,axiom,
! [F: a > a,G: a > a] :
( ( comp_l467081738950601185list_a @ ( map_a_a @ F ) @ ( map_a_a @ G ) )
= ( map_a_a @ ( comp_a_a_a @ F @ G ) ) ) ).
% map_comp_map
thf(fact_106_map__map,axiom,
! [F: extended_ereal > extended_ereal,G: a > extended_ereal,Xs: list_a] :
( ( map_Ex83193753010212676_ereal @ F @ ( map_a_Extended_ereal @ G @ Xs ) )
= ( map_a_Extended_ereal @ ( comp_E1870838029643375451real_a @ F @ G ) @ Xs ) ) ).
% map_map
thf(fact_107_map__map,axiom,
! [F: extended_ereal > a,G: a > extended_ereal,Xs: list_a] :
( ( map_Extended_ereal_a @ F @ ( map_a_Extended_ereal @ G @ Xs ) )
= ( map_a_a @ ( comp_E4980989167044428331al_a_a @ F @ G ) @ Xs ) ) ).
% map_map
thf(fact_108_map__map,axiom,
! [F: a > extended_ereal,G: a > a,Xs: list_a] :
( ( map_a_Extended_ereal @ F @ ( map_a_a @ G @ Xs ) )
= ( map_a_Extended_ereal @ ( comp_a6852332661818589707real_a @ F @ G ) @ Xs ) ) ).
% map_map
thf(fact_109_map__map,axiom,
! [F: a > a,G: a > a,Xs: list_a] :
( ( map_a_a @ F @ ( map_a_a @ G @ Xs ) )
= ( map_a_a @ ( comp_a_a_a @ F @ G ) @ Xs ) ) ).
% map_map
thf(fact_110_List_Omap_Ocompositionality,axiom,
! [F: extended_ereal > extended_ereal,G: a > extended_ereal,List: list_a] :
( ( map_Ex83193753010212676_ereal @ F @ ( map_a_Extended_ereal @ G @ List ) )
= ( map_a_Extended_ereal @ ( comp_E1870838029643375451real_a @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_111_List_Omap_Ocompositionality,axiom,
! [F: extended_ereal > a,G: a > extended_ereal,List: list_a] :
( ( map_Extended_ereal_a @ F @ ( map_a_Extended_ereal @ G @ List ) )
= ( map_a_a @ ( comp_E4980989167044428331al_a_a @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_112_List_Omap_Ocompositionality,axiom,
! [F: a > extended_ereal,G: a > a,List: list_a] :
( ( map_a_Extended_ereal @ F @ ( map_a_a @ G @ List ) )
= ( map_a_Extended_ereal @ ( comp_a6852332661818589707real_a @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_113_List_Omap_Ocompositionality,axiom,
! [F: a > a,G: a > a,List: list_a] :
( ( map_a_a @ F @ ( map_a_a @ G @ List ) )
= ( map_a_a @ ( comp_a_a_a @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_114_list_Omap__comp,axiom,
! [G: extended_ereal > extended_ereal,F: a > extended_ereal,V: list_a] :
( ( map_Ex83193753010212676_ereal @ G @ ( map_a_Extended_ereal @ F @ V ) )
= ( map_a_Extended_ereal @ ( comp_E1870838029643375451real_a @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_115_list_Omap__comp,axiom,
! [G: extended_ereal > a,F: a > extended_ereal,V: list_a] :
( ( map_Extended_ereal_a @ G @ ( map_a_Extended_ereal @ F @ V ) )
= ( map_a_a @ ( comp_E4980989167044428331al_a_a @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_116_list_Omap__comp,axiom,
! [G: a > extended_ereal,F: a > a,V: list_a] :
( ( map_a_Extended_ereal @ G @ ( map_a_a @ F @ V ) )
= ( map_a_Extended_ereal @ ( comp_a6852332661818589707real_a @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_117_list_Omap__comp,axiom,
! [G: a > a,F: a > a,V: list_a] :
( ( map_a_a @ G @ ( map_a_a @ F @ V ) )
= ( map_a_a @ ( comp_a_a_a @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_118_map__eq__conv,axiom,
! [F: a > extended_ereal,Xs: list_a,G: a > extended_ereal] :
( ( ( map_a_Extended_ereal @ F @ Xs )
= ( map_a_Extended_ereal @ G @ Xs ) )
= ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_119_map__eq__conv,axiom,
! [F: a > a,Xs: list_a,G: a > a] :
( ( ( map_a_a @ F @ Xs )
= ( map_a_a @ G @ Xs ) )
= ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_120_sum__list__eq__0__iff,axiom,
! [Ns: list_nat] :
( ( ( groups4561878855575611511st_nat @ Ns )
= zero_zero_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ns ) )
=> ( X = zero_zero_nat ) ) ) ) ).
% sum_list_eq_0_iff
thf(fact_121_char_Oexhaust,axiom,
! [Y: char] :
~ ! [X12: $o,X23: $o,X33: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( Y
!= ( char2 @ X12 @ X23 @ X33 @ X42 @ X52 @ X62 @ X72 @ X82 ) ) ).
% char.exhaust
thf(fact_122_add__right__imp__eq,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_123_add__right__imp__eq,axiom,
! [B: real,A: real,C2: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_124_add__right__imp__eq,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_125_add__left__imp__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_126_add__left__imp__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_127_add__left__imp__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_128_add_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C2 ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_129_add_Oleft__commute,axiom,
! [B: extended_ereal,A: extended_ereal,C2: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ B @ ( plus_p7876563987511257093_ereal @ A @ C2 ) )
= ( plus_p7876563987511257093_ereal @ A @ ( plus_p7876563987511257093_ereal @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_130_add_Oleft__commute,axiom,
! [B: real,A: real,C2: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C2 ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_131_add_Oleft__commute,axiom,
! [B: int,A: int,C2: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C2 ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_132_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_133_add_Ocommute,axiom,
( plus_p7876563987511257093_ereal
= ( ^ [A2: extended_ereal,B2: extended_ereal] : ( plus_p7876563987511257093_ereal @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_134_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A2: real,B2: real] : ( plus_plus_real @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_135_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A2: int,B2: int] : ( plus_plus_int @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_136_add_Oright__cancel,axiom,
! [B: real,A: real,C2: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C2 @ A ) )
= ( B = C2 ) ) ).
% add.right_cancel
thf(fact_137_add_Oright__cancel,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
= ( B = C2 ) ) ).
% add.right_cancel
thf(fact_138_add_Oleft__cancel,axiom,
! [A: real,B: real,C2: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C2 ) )
= ( B = C2 ) ) ).
% add.left_cancel
thf(fact_139_add_Oleft__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
= ( B = C2 ) ) ).
% add.left_cancel
thf(fact_140_add_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_141_add_Oassoc,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ ( plus_p7876563987511257093_ereal @ A @ B ) @ C2 )
= ( plus_p7876563987511257093_ereal @ A @ ( plus_p7876563987511257093_ereal @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_142_add_Oassoc,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_143_add_Oassoc,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_144_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_145_group__cancel_Oadd2,axiom,
! [B3: extended_ereal,K: extended_ereal,B: extended_ereal,A: extended_ereal] :
( ( B3
= ( plus_p7876563987511257093_ereal @ K @ B ) )
=> ( ( plus_p7876563987511257093_ereal @ A @ B3 )
= ( plus_p7876563987511257093_ereal @ K @ ( plus_p7876563987511257093_ereal @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_146_group__cancel_Oadd2,axiom,
! [B3: real,K: real,B: real,A: real] :
( ( B3
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B3 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_147_group__cancel_Oadd2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B3 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_148_group__cancel_Oadd1,axiom,
! [A3: nat,K: nat,A: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_149_group__cancel_Oadd1,axiom,
! [A3: extended_ereal,K: extended_ereal,A: extended_ereal,B: extended_ereal] :
( ( A3
= ( plus_p7876563987511257093_ereal @ K @ A ) )
=> ( ( plus_p7876563987511257093_ereal @ A3 @ B )
= ( plus_p7876563987511257093_ereal @ K @ ( plus_p7876563987511257093_ereal @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_150_group__cancel_Oadd1,axiom,
! [A3: real,K: real,A: real,B: real] :
( ( A3
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A3 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_151_group__cancel_Oadd1,axiom,
! [A3: int,K: int,A: int,B: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A3 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_152_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_153_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: extended_ereal,J: extended_ereal,K: extended_ereal,L: extended_ereal] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_p7876563987511257093_ereal @ I @ K )
= ( plus_p7876563987511257093_ereal @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_154_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_155_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_156_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_157_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ ( plus_p7876563987511257093_ereal @ A @ B ) @ C2 )
= ( plus_p7876563987511257093_ereal @ A @ ( plus_p7876563987511257093_ereal @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_158_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_159_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_160_char_Osize__gen,axiom,
! [X1: $o,X22: $o,X32: $o,X4: $o,X5: $o,X6: $o,X7: $o,X8: $o] :
( ( size_char @ ( char2 @ X1 @ X22 @ X32 @ X4 @ X5 @ X6 @ X7 @ X8 ) )
= zero_zero_nat ) ).
% char.size_gen
thf(fact_161_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_162_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_163_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_164_add_Ocomm__neutral,axiom,
! [A: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ A @ zero_z2744965634713055877_ereal )
= A ) ).
% add.comm_neutral
thf(fact_165_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_166_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_167_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_168_comm__monoid__add__class_Oadd__0,axiom,
! [A: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ zero_z2744965634713055877_ereal @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_169_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_170_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_171_ex__map__conv,axiom,
! [Ys: list_Extended_ereal,F: a > extended_ereal] :
( ( ? [Xs2: list_a] :
( Ys
= ( map_a_Extended_ereal @ F @ Xs2 ) ) )
= ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ ( set_Extended_ereal2 @ Ys ) )
=> ? [Y9: a] :
( X
= ( F @ Y9 ) ) ) ) ) ).
% ex_map_conv
thf(fact_172_ex__map__conv,axiom,
! [Ys: list_a,F: a > a] :
( ( ? [Xs2: list_a] :
( Ys
= ( map_a_a @ F @ Xs2 ) ) )
= ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Ys ) )
=> ? [Y9: a] :
( X
= ( F @ Y9 ) ) ) ) ) ).
% ex_map_conv
thf(fact_173_map__cong,axiom,
! [Xs: list_a,Ys: list_a,F: a > extended_ereal,G: a > extended_ereal] :
( ( Xs = Ys )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_a_Extended_ereal @ F @ Xs )
= ( map_a_Extended_ereal @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_174_map__cong,axiom,
! [Xs: list_a,Ys: list_a,F: a > a,G: a > a] :
( ( Xs = Ys )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_a_a @ F @ Xs )
= ( map_a_a @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_175_map__idI,axiom,
! [Xs: list_a,F: a > a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_a_a @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_176_map__ext,axiom,
! [Xs: list_a,F: a > extended_ereal,G: a > extended_ereal] :
( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_a_Extended_ereal @ F @ Xs )
= ( map_a_Extended_ereal @ G @ Xs ) ) ) ).
% map_ext
thf(fact_177_map__ext,axiom,
! [Xs: list_a,F: a > a,G: a > a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_a_a @ F @ Xs )
= ( map_a_a @ G @ Xs ) ) ) ).
% map_ext
thf(fact_178_list_Omap__ident__strong,axiom,
! [T: list_a,F: a > a] :
( ! [Z: a] :
( ( member_a @ Z @ ( set_a2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_a_a @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_179_list_Oinj__map__strong,axiom,
! [X2: list_a,Xa: list_a,F: a > extended_ereal,Fa: a > extended_ereal] :
( ! [Z: a,Za: a] :
( ( member_a @ Z @ ( set_a2 @ X2 ) )
=> ( ( member_a @ Za @ ( set_a2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_a_Extended_ereal @ F @ X2 )
= ( map_a_Extended_ereal @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_180_list_Oinj__map__strong,axiom,
! [X2: list_a,Xa: list_a,F: a > a,Fa: a > a] :
( ! [Z: a,Za: a] :
( ( member_a @ Z @ ( set_a2 @ X2 ) )
=> ( ( member_a @ Za @ ( set_a2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_a_a @ F @ X2 )
= ( map_a_a @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_181_list_Omap__cong0,axiom,
! [X2: list_a,F: a > extended_ereal,G: a > extended_ereal] :
( ! [Z: a] :
( ( member_a @ Z @ ( set_a2 @ X2 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_a_Extended_ereal @ F @ X2 )
= ( map_a_Extended_ereal @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_182_list_Omap__cong0,axiom,
! [X2: list_a,F: a > a,G: a > a] :
( ! [Z: a] :
( ( member_a @ Z @ ( set_a2 @ X2 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_a_a @ F @ X2 )
= ( map_a_a @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_183_list_Omap__cong,axiom,
! [X2: list_a,Ya: list_a,F: a > extended_ereal,G: a > extended_ereal] :
( ( X2 = Ya )
=> ( ! [Z: a] :
( ( member_a @ Z @ ( set_a2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_a_Extended_ereal @ F @ X2 )
= ( map_a_Extended_ereal @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_184_list_Omap__cong,axiom,
! [X2: list_a,Ya: list_a,F: a > a,G: a > a] :
( ( X2 = Ya )
=> ( ! [Z: a] :
( ( member_a @ Z @ ( set_a2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_a_a @ F @ X2 )
= ( map_a_a @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_185_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_186_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_187_list_Osimps_I8_J,axiom,
! [F: a > extended_ereal] :
( ( map_a_Extended_ereal @ F @ nil_a )
= nil_Extended_ereal ) ).
% list.simps(8)
thf(fact_188_list_Osimps_I8_J,axiom,
! [F: a > a] :
( ( map_a_a @ F @ nil_a )
= nil_a ) ).
% list.simps(8)
thf(fact_189_filter__map,axiom,
! [P: extended_ereal > $o,F: a > extended_ereal,Xs: list_a] :
( ( filter116404481290734027_ereal @ P @ ( map_a_Extended_ereal @ F @ Xs ) )
= ( map_a_Extended_ereal @ F @ ( filter_a @ ( comp_E512332703494744593al_o_a @ P @ F ) @ Xs ) ) ) ).
% filter_map
thf(fact_190_filter__map,axiom,
! [P: a > $o,F: a > a,Xs: list_a] :
( ( filter_a @ P @ ( map_a_a @ F @ Xs ) )
= ( map_a_a @ F @ ( filter_a @ ( comp_a_o_a @ P @ F ) @ Xs ) ) ) ).
% filter_map
thf(fact_191_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_192_list_Osize_I3_J,axiom,
( ( size_s2768339837476157504_ereal @ nil_Extended_ereal )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_193_sum__length__filter__compl,axiom,
! [P: a > $o,Xs: list_a] :
( ( plus_plus_nat @ ( size_size_list_a @ ( filter_a @ P @ Xs ) )
@ ( size_size_list_a
@ ( filter_a
@ ^ [X: a] :
~ ( P @ X )
@ Xs ) ) )
= ( size_size_list_a @ Xs ) ) ).
% sum_length_filter_compl
thf(fact_194_sum__length__filter__compl,axiom,
! [P: extended_ereal > $o,Xs: list_Extended_ereal] :
( ( plus_plus_nat @ ( size_s2768339837476157504_ereal @ ( filter116404481290734027_ereal @ P @ Xs ) )
@ ( size_s2768339837476157504_ereal
@ ( filter116404481290734027_ereal
@ ^ [X: extended_ereal] :
~ ( P @ X )
@ Xs ) ) )
= ( size_s2768339837476157504_ereal @ Xs ) ) ).
% sum_length_filter_compl
thf(fact_195_sum__list__addf,axiom,
! [F: a > extended_ereal,G: a > extended_ereal,Xs: list_a] :
( ( groups3567983573054521703_ereal
@ ( map_a_Extended_ereal
@ ^ [X: a] : ( plus_p7876563987511257093_ereal @ ( F @ X ) @ ( G @ X ) )
@ Xs ) )
= ( plus_p7876563987511257093_ereal @ ( groups3567983573054521703_ereal @ ( map_a_Extended_ereal @ F @ Xs ) ) @ ( groups3567983573054521703_ereal @ ( map_a_Extended_ereal @ G @ Xs ) ) ) ) ).
% sum_list_addf
thf(fact_196_double__eq__0__iff,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_197_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_198_lattice__ab__group__add__class_Odouble__zero,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% lattice_ab_group_add_class.double_zero
thf(fact_199_lattice__ab__group__add__class_Odouble__zero,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% lattice_ab_group_add_class.double_zero
thf(fact_200_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
= ( P @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_201_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_202_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_203_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_204_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_205_add__0__iff,axiom,
! [B: real,A: real] :
( ( B
= ( plus_plus_real @ B @ A ) )
= ( A = zero_zero_real ) ) ).
% add_0_iff
thf(fact_206_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_207_sum__list__map__remove1,axiom,
! [X2: a,Xs: list_a,F: a > extended_ereal] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ( groups3567983573054521703_ereal @ ( map_a_Extended_ereal @ F @ Xs ) )
= ( plus_p7876563987511257093_ereal @ ( F @ X2 ) @ ( groups3567983573054521703_ereal @ ( map_a_Extended_ereal @ F @ ( remove1_a @ X2 @ Xs ) ) ) ) ) ) ).
% sum_list_map_remove1
thf(fact_208_sum__list__def,axiom,
( groups4561878855575611511st_nat
= ( groups_monoid_F_nat @ plus_plus_nat @ zero_zero_nat ) ) ).
% sum_list_def
thf(fact_209_sum__list__def,axiom,
( groups6723090944982001619t_real
= ( groups_monoid_F_real @ plus_plus_real @ zero_zero_real ) ) ).
% sum_list_def
thf(fact_210_sum__list__def,axiom,
( groups4559388385066561235st_int
= ( groups_monoid_F_int @ plus_plus_int @ zero_zero_int ) ) ).
% sum_list_def
thf(fact_211_sum__list__def,axiom,
( groups3567983573054521703_ereal
= ( groups4041799636089507985_ereal @ plus_p7876563987511257093_ereal @ zero_z2744965634713055877_ereal ) ) ).
% sum_list_def
thf(fact_212_ereal__PInfty__eq__plus,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( extend1530274965995635425_ereal
= ( plus_p7876563987511257093_ereal @ A @ B ) )
= ( ( A = extend1530274965995635425_ereal )
| ( B = extend1530274965995635425_ereal ) ) ) ).
% ereal_PInfty_eq_plus
thf(fact_213_ereal__plus__eq__PInfty,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( plus_p7876563987511257093_ereal @ A @ B )
= extend1530274965995635425_ereal )
= ( ( A = extend1530274965995635425_ereal )
| ( B = extend1530274965995635425_ereal ) ) ) ).
% ereal_plus_eq_PInfty
thf(fact_214_ereal__0__plus,axiom,
! [X2: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ ( extended_ereal2 @ zero_zero_real ) @ X2 )
= X2 ) ).
% ereal_0_plus
thf(fact_215_plus__ereal__0,axiom,
! [X2: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ X2 @ ( extended_ereal2 @ zero_zero_real ) )
= X2 ) ).
% plus_ereal_0
thf(fact_216_plus__ereal_Osimps_I1_J,axiom,
! [R: real,P2: real] :
( ( plus_p7876563987511257093_ereal @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P2 ) )
= ( extended_ereal2 @ ( plus_plus_real @ R @ P2 ) ) ) ).
% plus_ereal.simps(1)
thf(fact_217_plus__ereal_Osimps_I3_J,axiom,
! [A: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ A @ extend1530274965995635425_ereal )
= extend1530274965995635425_ereal ) ).
% plus_ereal.simps(3)
thf(fact_218_plus__ereal_Osimps_I2_J,axiom,
! [A: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ extend1530274965995635425_ereal @ A )
= extend1530274965995635425_ereal ) ).
% plus_ereal.simps(2)
thf(fact_219_ereal_Osize_I6_J,axiom,
( ( size_s5012395470859603514_ereal @ extended_MInfty )
= zero_zero_nat ) ).
% ereal.size(6)
thf(fact_220_ereal_Osize_I5_J,axiom,
( ( size_s5012395470859603514_ereal @ extended_PInfty )
= zero_zero_nat ) ).
% ereal.size(5)
thf(fact_221_ereal_Osize__gen_I3_J,axiom,
( ( extended_size_ereal @ extended_MInfty )
= zero_zero_nat ) ).
% ereal.size_gen(3)
thf(fact_222_ereal_Osize__gen_I2_J,axiom,
( ( extended_size_ereal @ extended_PInfty )
= zero_zero_nat ) ).
% ereal.size_gen(2)
thf(fact_223_length__splice,axiom,
! [Xs: list_a,Ys: list_a] :
( ( size_size_list_a @ ( splice_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) ) ) ).
% length_splice
thf(fact_224_length__splice,axiom,
! [Xs: list_Extended_ereal,Ys: list_Extended_ereal] :
( ( size_s2768339837476157504_ereal @ ( splice291902714325861031_ereal @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_s2768339837476157504_ereal @ Xs ) @ ( size_s2768339837476157504_ereal @ Ys ) ) ) ).
% length_splice
thf(fact_225_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_226_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_227_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_228_ereal__less__PInfty,axiom,
! [A: extended_ereal] :
( ( A != extend1530274965995635425_ereal )
=> ( ord_le1188267648640031866_ereal @ A @ extend1530274965995635425_ereal ) ) ).
% ereal_less_PInfty
thf(fact_229_ereal__infty__less_I1_J,axiom,
! [X2: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X2 @ extend1530274965995635425_ereal )
= ( X2 != extend1530274965995635425_ereal ) ) ).
% ereal_infty_less(1)
thf(fact_230_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_231_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_232_add__less__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_233_add__less__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_234_add__less__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_235_add__less__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_236_add__less__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_237_add__less__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_238_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_239_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_240_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_241_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_242_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_243_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_244_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_245_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_246_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_247_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_248_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_249_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_250_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_251_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_252_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_253_linordered__ab__group__add__class_Ozero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% linordered_ab_group_add_class.zero_less_double_add_iff_zero_less_single_add
thf(fact_254_linordered__ab__group__add__class_Ozero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% linordered_ab_group_add_class.zero_less_double_add_iff_zero_less_single_add
thf(fact_255_double__add__less__zero__iff__single__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_less_zero
thf(fact_256_double__add__less__zero__iff__single__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_less_zero
thf(fact_257_lattice__ab__group__add__class_Ozero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% lattice_ab_group_add_class.zero_less_double_add_iff_zero_less_single_add
thf(fact_258_lattice__ab__group__add__class_Ozero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% lattice_ab_group_add_class.zero_less_double_add_iff_zero_less_single_add
thf(fact_259_length__greater__0__conv,axiom,
! [Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) )
= ( Xs != nil_a ) ) ).
% length_greater_0_conv
thf(fact_260_length__greater__0__conv,axiom,
! [Xs: list_Extended_ereal] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s2768339837476157504_ereal @ Xs ) )
= ( Xs != nil_Extended_ereal ) ) ).
% length_greater_0_conv
thf(fact_261_ereal__less_I1_J,axiom,
! [R: real] :
( ( ord_le1188267648640031866_ereal @ ( extended_ereal2 @ R ) @ zero_z2744965634713055877_ereal )
= ( ord_less_real @ R @ zero_zero_real ) ) ).
% ereal_less(1)
thf(fact_262_ereal__less_I2_J,axiom,
! [R: real] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ ( extended_ereal2 @ R ) )
= ( ord_less_real @ zero_zero_real @ R ) ) ).
% ereal_less(2)
thf(fact_263_ereal__add__strict__mono2,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal,D: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ( ord_le1188267648640031866_ereal @ C2 @ D )
=> ( ord_le1188267648640031866_ereal @ ( plus_p7876563987511257093_ereal @ A @ C2 ) @ ( plus_p7876563987511257093_ereal @ B @ D ) ) ) ) ).
% ereal_add_strict_mono2
thf(fact_264_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_265_verit__comp__simplify1_I1_J,axiom,
! [A: extended_ereal] :
~ ( ord_le1188267648640031866_ereal @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_266_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_267_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_268_less__ereal_Osimps_I1_J,axiom,
! [X2: real,Y: real] :
( ( ord_le1188267648640031866_ereal @ ( extended_ereal2 @ X2 ) @ ( extended_ereal2 @ Y ) )
= ( ord_less_real @ X2 @ Y ) ) ).
% less_ereal.simps(1)
thf(fact_269_ereal_Odistinct_I5_J,axiom,
extended_PInfty != extended_MInfty ).
% ereal.distinct(5)
thf(fact_270_ereal_Oexhaust,axiom,
! [Y: extended_ereal] :
( ! [X12: real] :
( Y
!= ( extended_ereal2 @ X12 ) )
=> ( ( Y != extended_PInfty )
=> ( Y = extended_MInfty ) ) ) ).
% ereal.exhaust
thf(fact_271_uminus__ereal_Ocases,axiom,
! [X2: extended_ereal] :
( ! [R2: real] :
( X2
!= ( extended_ereal2 @ R2 ) )
=> ( ( X2 != extended_PInfty )
=> ( X2 = extended_MInfty ) ) ) ).
% uminus_ereal.cases
thf(fact_272_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_273_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_274_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_275_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_276_add__less__imp__less__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_277_add__less__imp__less__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_278_add__less__imp__less__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_279_add__less__imp__less__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_280_add__less__imp__less__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_281_add__less__imp__less__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_282_add__strict__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_283_add__strict__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_284_add__strict__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_285_add__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_286_add__strict__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_287_add__strict__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_288_add__strict__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_289_add__strict__mono,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C2 @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_290_add__strict__mono,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_291_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_292_add__mono__thms__linordered__field_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_293_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_294_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_295_add__mono__thms__linordered__field_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_296_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_297_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_298_add__mono__thms__linordered__field_I5_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_299_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_300_less__ereal_Osimps_I2_J,axiom,
! [A: extended_ereal] :
~ ( ord_le1188267648640031866_ereal @ extend1530274965995635425_ereal @ A ) ).
% less_ereal.simps(2)
thf(fact_301_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_302_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_303_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_304_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_305_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_306_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_307_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_308_ereal__dense2,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X2 @ Y )
=> ? [Z: real] :
( ( ord_le1188267648640031866_ereal @ X2 @ ( extended_ereal2 @ Z ) )
& ( ord_le1188267648640031866_ereal @ ( extended_ereal2 @ Z ) @ Y ) ) ) ).
% ereal_dense2
thf(fact_309_length__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ! [Xs3: list_a] :
( ! [Ys2: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys2 ) @ ( size_size_list_a @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_310_length__induct,axiom,
! [P: list_Extended_ereal > $o,Xs: list_Extended_ereal] :
( ! [Xs3: list_Extended_ereal] :
( ! [Ys2: list_Extended_ereal] :
( ( ord_less_nat @ ( size_s2768339837476157504_ereal @ Ys2 ) @ ( size_s2768339837476157504_ereal @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_311_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_312_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_313_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_314_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_315_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_316_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_317_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_318_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_319_infinity__ereal__def,axiom,
extend1530274965995635425_ereal = extended_PInfty ).
% infinity_ereal_def
thf(fact_320_ereal_Odistinct_I3_J,axiom,
! [X1: real] :
( ( extended_ereal2 @ X1 )
!= extended_MInfty ) ).
% ereal.distinct(3)
thf(fact_321_ereal_Odistinct_I1_J,axiom,
! [X1: real] :
( ( extended_ereal2 @ X1 )
!= extended_PInfty ) ).
% ereal.distinct(1)
thf(fact_322_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_323_add__neg__neg,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ zero_z2744965634713055877_ereal )
=> ( ( ord_le1188267648640031866_ereal @ B @ zero_z2744965634713055877_ereal )
=> ( ord_le1188267648640031866_ereal @ ( plus_p7876563987511257093_ereal @ A @ B ) @ zero_z2744965634713055877_ereal ) ) ) ).
% add_neg_neg
thf(fact_324_add__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_325_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_326_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_327_add__pos__pos,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ B )
=> ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ ( plus_p7876563987511257093_ereal @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_328_add__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_329_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_330_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_331_pos__add__strict,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_332_pos__add__strict,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_333_pos__add__strict,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_334_less__ereal_Osimps_I4_J,axiom,
! [X2: real] : ( ord_le1188267648640031866_ereal @ ( extended_ereal2 @ X2 ) @ extend1530274965995635425_ereal ) ).
% less_ereal.simps(4)
thf(fact_335_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_336_ereal__less_I5_J,axiom,
ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ extend1530274965995635425_ereal ).
% ereal_less(5)
thf(fact_337_length__pos__if__in__set,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_338_length__pos__if__in__set,axiom,
! [X2: extended_ereal,Xs: list_Extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ ( set_Extended_ereal2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s2768339837476157504_ereal @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_339_length__filter__less,axiom,
! [X2: a,Xs: list_a,P: a > $o] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ~ ( P @ X2 )
=> ( ord_less_nat @ ( size_size_list_a @ ( filter_a @ P @ Xs ) ) @ ( size_size_list_a @ Xs ) ) ) ) ).
% length_filter_less
thf(fact_340_length__filter__less,axiom,
! [X2: extended_ereal,Xs: list_Extended_ereal,P: extended_ereal > $o] :
( ( member2350847679896131959_ereal @ X2 @ ( set_Extended_ereal2 @ Xs ) )
=> ( ~ ( P @ X2 )
=> ( ord_less_nat @ ( size_s2768339837476157504_ereal @ ( filter116404481290734027_ereal @ P @ Xs ) ) @ ( size_s2768339837476157504_ereal @ Xs ) ) ) ) ).
% length_filter_less
thf(fact_341_add__less__zeroD,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X2 @ Y ) @ zero_zero_real )
=> ( ( ord_less_real @ X2 @ zero_zero_real )
| ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_342_add__less__zeroD,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X2 @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X2 @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_343_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_344_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_345_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_346_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_347_plus__ereal_Oelims,axiom,
! [X2: extended_ereal,Xa: extended_ereal,Y: extended_ereal] :
( ( ( plus_p7876563987511257093_ereal @ X2 @ Xa )
= Y )
=> ( ! [R2: real] :
( ( X2
= ( extended_ereal2 @ R2 ) )
=> ! [P3: real] :
( ( Xa
= ( extended_ereal2 @ P3 ) )
=> ( Y
!= ( extended_ereal2 @ ( plus_plus_real @ R2 @ P3 ) ) ) ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ( Y != extend1530274965995635425_ereal ) )
=> ( ( ( Xa = extend1530274965995635425_ereal )
=> ( Y != extend1530274965995635425_ereal ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Y
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ? [P3: real] :
( Xa
= ( extended_ereal2 @ P3 ) )
=> ( Y
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) )
=> ~ ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( Xa
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Y
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ) ) ) ) ) ).
% plus_ereal.elims
thf(fact_348_bit__count__append,axiom,
! [X2: option_list_o,Y: option_list_o] :
( ( prefix3213528784805800034_count @ ( prefix5314359684614007693append @ X2 @ Y ) )
= ( plus_p7876563987511257093_ereal @ ( prefix3213528784805800034_count @ X2 ) @ ( prefix3213528784805800034_count @ Y ) ) ) ).
% bit_count_append
thf(fact_349_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_350_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_351_neg__equal__iff__equal,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_352_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_353_verit__minus__simplify_I4_J,axiom,
! [B: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_354_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_355_ereal__uminus__eq__iff,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( uminus27091377158695749_ereal @ A )
= ( uminus27091377158695749_ereal @ B ) )
= ( A = B ) ) ).
% ereal_uminus_eq_iff
thf(fact_356_ereal__uminus__uminus,axiom,
! [A: extended_ereal] :
( ( uminus27091377158695749_ereal @ ( uminus27091377158695749_ereal @ A ) )
= A ) ).
% ereal_uminus_uminus
thf(fact_357_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_358_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_359_neg__0__equal__iff__equal,axiom,
! [A: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A ) )
= ( zero_zero_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_360_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_361_neg__equal__0__iff__equal,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_362_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_363_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_364_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_365_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_366_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_367_neg__less__iff__less,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_368_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_369_add__minus__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_370_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_371_minus__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_372_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_373_minus__add__distrib,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% minus_add_distrib
thf(fact_374_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_375_ereal__uminus__zero,axiom,
( ( uminus27091377158695749_ereal @ zero_z2744965634713055877_ereal )
= zero_z2744965634713055877_ereal ) ).
% ereal_uminus_zero
thf(fact_376_ereal__uminus__zero__iff,axiom,
! [A: extended_ereal] :
( ( ( uminus27091377158695749_ereal @ A )
= zero_z2744965634713055877_ereal )
= ( A = zero_z2744965634713055877_ereal ) ) ).
% ereal_uminus_zero_iff
thf(fact_377_ereal__minus__less__minus,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ A ) @ ( uminus27091377158695749_ereal @ B ) )
= ( ord_le1188267648640031866_ereal @ B @ A ) ) ).
% ereal_minus_less_minus
thf(fact_378_less__neg__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_379_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_380_neg__less__pos,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_pos
thf(fact_381_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_382_neg__0__less__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_383_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_384_neg__less__0__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_0_iff_less
thf(fact_385_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_386_ab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_left_minus
thf(fact_387_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_388_add_Oright__inverse,axiom,
! [A: real] :
( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
= zero_zero_real ) ).
% add.right_inverse
thf(fact_389_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_390_ereal__infty__less_I2_J,axiom,
! [X2: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ X2 )
= ( X2
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ).
% ereal_infty_less(2)
thf(fact_391_ereal__MInfty__lessI,axiom,
! [A: extended_ereal] :
( ( A
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ A ) ) ).
% ereal_MInfty_lessI
thf(fact_392_ereal__plus__eq__MInfty,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( plus_p7876563987511257093_ereal @ A @ B )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
= ( ( ( A
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
| ( B
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) )
& ( A != extend1530274965995635425_ereal )
& ( B != extend1530274965995635425_ereal ) ) ) ).
% ereal_plus_eq_MInfty
thf(fact_393_ereal__MInfty__eq__plus,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal )
= ( plus_p7876563987511257093_ereal @ A @ B ) )
= ( ( ( A
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
& ( B != extend1530274965995635425_ereal ) )
| ( ( B
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
& ( A != extend1530274965995635425_ereal ) ) ) ) ).
% ereal_MInfty_eq_plus
thf(fact_394_neg__0__less__iff__less__erea,axiom,
! [A: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ ( uminus27091377158695749_ereal @ A ) )
= ( ord_le1188267648640031866_ereal @ A @ zero_z2744965634713055877_ereal ) ) ).
% neg_0_less_iff_less_erea
thf(fact_395_MInfty__eq__minfinity,axiom,
( extended_MInfty
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% MInfty_eq_minfinity
thf(fact_396_is__num__normalize_I8_J,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_397_is__num__normalize_I8_J,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_398_ereal__uminus__less__reorder,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ A ) @ B )
= ( ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ B ) @ A ) ) ).
% ereal_uminus_less_reorder
thf(fact_399_ereal__less__uminus__reorder,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ ( uminus27091377158695749_ereal @ B ) )
= ( ord_le1188267648640031866_ereal @ B @ ( uminus27091377158695749_ereal @ A ) ) ) ).
% ereal_less_uminus_reorder
thf(fact_400_linorder__neqE__nat,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_401_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_402_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_403_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_404_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_405_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_406_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_407_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_408_equation__minus__iff,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_409_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_410_minus__equation__iff,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( uminus_uminus_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_411_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_412_ereal__uminus__eq__reorder,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( uminus27091377158695749_ereal @ A )
= B )
= ( A
= ( uminus27091377158695749_ereal @ B ) ) ) ).
% ereal_uminus_eq_reorder
thf(fact_413_verit__negate__coefficient_I3_J,axiom,
! [A: real,B: real] :
( ( A = B )
=> ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_414_verit__negate__coefficient_I3_J,axiom,
! [A: int,B: int] :
( ( A = B )
=> ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_415_less__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% less_minus_iff
thf(fact_416_less__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_417_minus__less__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_418_minus__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_419_verit__negate__coefficient_I2_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_420_verit__negate__coefficient_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_421_group__cancel_Oneg1,axiom,
! [A3: real,K: real,A: real] :
( ( A3
= ( plus_plus_real @ K @ A ) )
=> ( ( uminus_uminus_real @ A3 )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_422_group__cancel_Oneg1,axiom,
! [A3: int,K: int,A: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( uminus_uminus_int @ A3 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_423_add_Oinverse__distrib__swap,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_424_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_425_MInfty__neq__PInfty_I1_J,axiom,
( extend1530274965995635425_ereal
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% MInfty_neq_PInfty(1)
thf(fact_426_less__ereal_Osimps_I6_J,axiom,
ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ extend1530274965995635425_ereal ).
% less_ereal.simps(6)
thf(fact_427_less__ereal_Osimps_I3_J,axiom,
! [A: extended_ereal] :
~ ( ord_le1188267648640031866_ereal @ A @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% less_ereal.simps(3)
thf(fact_428_neg__eq__iff__add__eq__0,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( plus_plus_real @ A @ B )
= zero_zero_real ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_429_neg__eq__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_430_eq__neg__iff__add__eq__0,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( ( plus_plus_real @ A @ B )
= zero_zero_real ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_431_eq__neg__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_432_add_Oinverse__unique,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= zero_zero_real )
=> ( ( uminus_uminus_real @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_433_add_Oinverse__unique,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_434_ab__group__add__class_Oab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_group_add_class.ab_left_minus
thf(fact_435_ab__group__add__class_Oab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_436_add__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= zero_zero_real )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% add_eq_0_iff
thf(fact_437_add__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% add_eq_0_iff
thf(fact_438_ereal__all__split,axiom,
( ( ^ [P4: extended_ereal > $o] :
! [X9: extended_ereal] : ( P4 @ X9 ) )
= ( ^ [P5: extended_ereal > $o] :
( ( P5 @ extend1530274965995635425_ereal )
& ! [X: real] : ( P5 @ ( extended_ereal2 @ X ) )
& ( P5 @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ).
% ereal_all_split
thf(fact_439_ereal__ex__split,axiom,
( ( ^ [P4: extended_ereal > $o] :
? [X9: extended_ereal] : ( P4 @ X9 ) )
= ( ^ [P5: extended_ereal > $o] :
( ( P5 @ extend1530274965995635425_ereal )
| ? [X: real] : ( P5 @ ( extended_ereal2 @ X ) )
| ( P5 @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ).
% ereal_ex_split
thf(fact_440_ereal3__cases,axiom,
! [X2: extended_ereal,Xa: extended_ereal,Xb: extended_ereal] :
( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ? [Ra: real] :
( Xa
= ( extended_ereal2 @ Ra ) )
=> ! [Rb: real] :
( Xb
!= ( extended_ereal2 @ Rb ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ? [Ra: real] :
( Xa
= ( extended_ereal2 @ Ra ) )
=> ( Xb != extend1530274965995635425_ereal ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ? [Ra: real] :
( Xa
= ( extended_ereal2 @ Ra ) )
=> ( Xb
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa = extend1530274965995635425_ereal )
=> ! [Ra: real] :
( Xb
!= ( extended_ereal2 @ Ra ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa = extend1530274965995635425_ereal )
=> ( Xb != extend1530274965995635425_ereal ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa = extend1530274965995635425_ereal )
=> ( Xb
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ! [Ra: real] :
( Xb
!= ( extended_ereal2 @ Ra ) ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Xb != extend1530274965995635425_ereal ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Xb
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ( ? [R2: real] :
( Xa
= ( extended_ereal2 @ R2 ) )
=> ! [Ra: real] :
( Xb
!= ( extended_ereal2 @ Ra ) ) ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ( ? [R2: real] :
( Xa
= ( extended_ereal2 @ R2 ) )
=> ( Xb != extend1530274965995635425_ereal ) ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ( ? [R2: real] :
( Xa
= ( extended_ereal2 @ R2 ) )
=> ( Xb
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ( ( Xa = extend1530274965995635425_ereal )
=> ! [R2: real] :
( Xb
!= ( extended_ereal2 @ R2 ) ) ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ( ( Xa = extend1530274965995635425_ereal )
=> ( Xb != extend1530274965995635425_ereal ) ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ( ( Xa = extend1530274965995635425_ereal )
=> ( Xb
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ( ( Xa
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ! [R2: real] :
( Xb
!= ( extended_ereal2 @ R2 ) ) ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ( ( Xa
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Xb != extend1530274965995635425_ereal ) ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ( ( Xa
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Xb
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ? [R2: real] :
( Xa
= ( extended_ereal2 @ R2 ) )
=> ! [Ra: real] :
( Xb
!= ( extended_ereal2 @ Ra ) ) ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ? [R2: real] :
( Xa
= ( extended_ereal2 @ R2 ) )
=> ( Xb != extend1530274965995635425_ereal ) ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ? [R2: real] :
( Xa
= ( extended_ereal2 @ R2 ) )
=> ( Xb
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( Xa = extend1530274965995635425_ereal )
=> ! [R2: real] :
( Xb
!= ( extended_ereal2 @ R2 ) ) ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( Xa = extend1530274965995635425_ereal )
=> ( Xb != extend1530274965995635425_ereal ) ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( Xa = extend1530274965995635425_ereal )
=> ( Xb
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( Xa
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ! [R2: real] :
( Xb
!= ( extended_ereal2 @ R2 ) ) ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( Xa
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Xb != extend1530274965995635425_ereal ) ) )
=> ~ ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( Xa
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Xb
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% ereal3_cases
thf(fact_441_ereal2__cases,axiom,
! [X2: extended_ereal,Xa: extended_ereal] :
( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ! [Ra: real] :
( Xa
!= ( extended_ereal2 @ Ra ) ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( Xa != extend1530274965995635425_ereal ) )
=> ( ( ? [R2: real] :
( X2
= ( extended_ereal2 @ R2 ) )
=> ( Xa
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ! [R2: real] :
( Xa
!= ( extended_ereal2 @ R2 ) ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ( Xa != extend1530274965995635425_ereal ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ( Xa
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ! [R2: real] :
( Xa
!= ( extended_ereal2 @ R2 ) ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Xa != extend1530274965995635425_ereal ) )
=> ~ ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Xa
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ) ) ) ) ) ) ).
% ereal2_cases
thf(fact_442_ereal__cases,axiom,
! [X2: extended_ereal] :
( ! [R2: real] :
( X2
!= ( extended_ereal2 @ R2 ) )
=> ( ( X2 != extend1530274965995635425_ereal )
=> ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ).
% ereal_cases
thf(fact_443_real__of__ereal_Ocases,axiom,
! [X2: extended_ereal] :
( ! [R2: real] :
( X2
!= ( extended_ereal2 @ R2 ) )
=> ( ( X2 != extend1530274965995635425_ereal )
=> ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ).
% real_of_ereal.cases
thf(fact_444_abs__ereal_Ocases,axiom,
! [X2: extended_ereal] :
( ! [R2: real] :
( X2
!= ( extended_ereal2 @ R2 ) )
=> ( ( X2
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( X2 = extend1530274965995635425_ereal ) ) ) ).
% abs_ereal.cases
thf(fact_445_MInfty__neq__ereal_I1_J,axiom,
! [R: real] :
( ( extended_ereal2 @ R )
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% MInfty_neq_ereal(1)
thf(fact_446_less__ereal_Osimps_I5_J,axiom,
! [R: real] : ( ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ ( extended_ereal2 @ R ) ) ).
% less_ereal.simps(5)
thf(fact_447_ereal__less__ereal__Ex,axiom,
! [X2: extended_ereal,R: real] :
( ( ord_le1188267648640031866_ereal @ X2 @ ( extended_ereal2 @ R ) )
= ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
| ? [P6: real] :
( ( ord_less_real @ P6 @ R )
& ( X2
= ( extended_ereal2 @ P6 ) ) ) ) ) ).
% ereal_less_ereal_Ex
thf(fact_448_less__ereal_Oelims_I1_J,axiom,
! [X2: extended_ereal,Xa: extended_ereal,Y: $o] :
( ( ( ord_le1188267648640031866_ereal @ X2 @ Xa )
= Y )
=> ( ! [X3: real] :
( ( X2
= ( extended_ereal2 @ X3 ) )
=> ! [Y10: real] :
( ( Xa
= ( extended_ereal2 @ Y10 ) )
=> ( Y
= ( ~ ( ord_less_real @ X3 @ Y10 ) ) ) ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> Y )
=> ( ( ( Xa
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> Y )
=> ( ( ? [X3: real] :
( X2
= ( extended_ereal2 @ X3 ) )
=> ( ( Xa = extend1530274965995635425_ereal )
=> ~ Y ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ? [R2: real] :
( Xa
= ( extended_ereal2 @ R2 ) )
=> ~ Y ) )
=> ~ ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( Xa = extend1530274965995635425_ereal )
=> ~ Y ) ) ) ) ) ) ) ) ).
% less_ereal.elims(1)
thf(fact_449_less__ereal_Oelims_I2_J,axiom,
! [X2: extended_ereal,Xa: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X2 @ Xa )
=> ( ! [X3: real] :
( ( X2
= ( extended_ereal2 @ X3 ) )
=> ! [Y10: real] :
( ( Xa
= ( extended_ereal2 @ Y10 ) )
=> ~ ( ord_less_real @ X3 @ Y10 ) ) )
=> ( ( ? [X3: real] :
( X2
= ( extended_ereal2 @ X3 ) )
=> ( Xa != extend1530274965995635425_ereal ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ! [R2: real] :
( Xa
!= ( extended_ereal2 @ R2 ) ) )
=> ~ ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Xa != extend1530274965995635425_ereal ) ) ) ) ) ) ).
% less_ereal.elims(2)
thf(fact_450_less__ereal_Oelims_I3_J,axiom,
! [X2: extended_ereal,Xa: extended_ereal] :
( ~ ( ord_le1188267648640031866_ereal @ X2 @ Xa )
=> ( ! [X3: real] :
( ( X2
= ( extended_ereal2 @ X3 ) )
=> ! [Y10: real] :
( ( Xa
= ( extended_ereal2 @ Y10 ) )
=> ( ord_less_real @ X3 @ Y10 ) ) )
=> ( ( X2 != extend1530274965995635425_ereal )
=> ( Xa
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ).
% less_ereal.elims(3)
thf(fact_451_Infty__neq__0_I3_J,axiom,
( ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal )
!= zero_z2744965634713055877_ereal ) ).
% Infty_neq_0(3)
thf(fact_452_ereal__less_I6_J,axiom,
ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ zero_z2744965634713055877_ereal ).
% ereal_less(6)
thf(fact_453_ereal__add__cancel__right,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( A
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( ( plus_p7876563987511257093_ereal @ B @ A )
= ( plus_p7876563987511257093_ereal @ C2 @ A ) )
= ( ( A = extend1530274965995635425_ereal )
| ( B = C2 ) ) ) ) ).
% ereal_add_cancel_right
thf(fact_454_ereal__add__cancel__left,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( A
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( ( plus_p7876563987511257093_ereal @ A @ B )
= ( plus_p7876563987511257093_ereal @ A @ C2 ) )
= ( ( A = extend1530274965995635425_ereal )
| ( B = C2 ) ) ) ) ).
% ereal_add_cancel_left
thf(fact_455_plus__ereal_Osimps_I6_J,axiom,
( ( plus_p7876563987511257093_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% plus_ereal.simps(6)
thf(fact_456_uminus__ereal_Osimps_I3_J,axiom,
( ( uminus27091377158695749_ereal @ extended_MInfty )
= extended_PInfty ) ).
% uminus_ereal.simps(3)
thf(fact_457_uminus__ereal_Osimps_I2_J,axiom,
( ( uminus27091377158695749_ereal @ extended_PInfty )
= extended_MInfty ) ).
% uminus_ereal.simps(2)
thf(fact_458_linorder__neqE__linordered__idom,axiom,
! [X2: real,Y: real] :
( ( X2 != Y )
=> ( ~ ( ord_less_real @ X2 @ Y )
=> ( ord_less_real @ Y @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_459_linorder__neqE__linordered__idom,axiom,
! [X2: int,Y: int] :
( ( X2 != Y )
=> ( ~ ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_460_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% is_num_normalize(1)
thf(fact_461_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% is_num_normalize(1)
thf(fact_462_plus__ereal_Osimps_I4_J,axiom,
! [R: real] :
( ( plus_p7876563987511257093_ereal @ ( extended_ereal2 @ R ) @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% plus_ereal.simps(4)
thf(fact_463_plus__ereal_Osimps_I5_J,axiom,
! [P2: real] :
( ( plus_p7876563987511257093_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ ( extended_ereal2 @ P2 ) )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% plus_ereal.simps(5)
thf(fact_464_times__ereal_Osimps_I5_J,axiom,
! [R: real] :
( ( ( R = zero_zero_real )
=> ( ( times_7703590493115627913_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ ( extended_ereal2 @ R ) )
= zero_z2744965634713055877_ereal ) )
& ( ( R != zero_zero_real )
=> ( ( ( ord_less_real @ zero_zero_real @ R )
=> ( ( times_7703590493115627913_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ ( extended_ereal2 @ R ) )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ R )
=> ( ( times_7703590493115627913_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ ( extended_ereal2 @ R ) )
= extend1530274965995635425_ereal ) ) ) ) ) ).
% times_ereal.simps(5)
thf(fact_465_times__ereal_Osimps_I4_J,axiom,
! [R: real] :
( ( ( R = zero_zero_real )
=> ( ( times_7703590493115627913_ereal @ ( extended_ereal2 @ R ) @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
= zero_z2744965634713055877_ereal ) )
& ( ( R != zero_zero_real )
=> ( ( ( ord_less_real @ zero_zero_real @ R )
=> ( ( times_7703590493115627913_ereal @ ( extended_ereal2 @ R ) @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ R )
=> ( ( times_7703590493115627913_ereal @ ( extended_ereal2 @ R ) @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
= extend1530274965995635425_ereal ) ) ) ) ) ).
% times_ereal.simps(4)
thf(fact_466_times__ereal_Osimps_I3_J,axiom,
! [R: real] :
( ( ( R = zero_zero_real )
=> ( ( times_7703590493115627913_ereal @ extend1530274965995635425_ereal @ ( extended_ereal2 @ R ) )
= zero_z2744965634713055877_ereal ) )
& ( ( R != zero_zero_real )
=> ( ( ( ord_less_real @ zero_zero_real @ R )
=> ( ( times_7703590493115627913_ereal @ extend1530274965995635425_ereal @ ( extended_ereal2 @ R ) )
= extend1530274965995635425_ereal ) )
& ( ~ ( ord_less_real @ zero_zero_real @ R )
=> ( ( times_7703590493115627913_ereal @ extend1530274965995635425_ereal @ ( extended_ereal2 @ R ) )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ) ).
% times_ereal.simps(3)
thf(fact_467_mult__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ( times_times_real @ A @ C2 )
= ( times_times_real @ B @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_468_mult__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_469_mult__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ( times_times_int @ A @ C2 )
= ( times_times_int @ B @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_470_mult__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ( times_times_real @ C2 @ A )
= ( times_times_real @ C2 @ B ) )
= ( ( C2 = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_471_mult__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B ) )
= ( ( C2 = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_472_mult__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ( times_times_int @ C2 @ A )
= ( times_times_int @ C2 @ B ) )
= ( ( C2 = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_473_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_474_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_475_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_476_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_477_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_478_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_479_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_480_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_481_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_482_real__add__minus__iff,axiom,
! [X2: real,A: real] :
( ( ( plus_plus_real @ X2 @ ( uminus_uminus_real @ A ) )
= zero_zero_real )
= ( X2 = A ) ) ).
% real_add_minus_iff
thf(fact_483_ereal__mult__minus__right,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( times_7703590493115627913_ereal @ A @ ( uminus27091377158695749_ereal @ B ) )
= ( uminus27091377158695749_ereal @ ( times_7703590493115627913_ereal @ A @ B ) ) ) ).
% ereal_mult_minus_right
thf(fact_484_ereal__mult__minus__left,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( times_7703590493115627913_ereal @ ( uminus27091377158695749_ereal @ A ) @ B )
= ( uminus27091377158695749_ereal @ ( times_7703590493115627913_ereal @ A @ B ) ) ) ).
% ereal_mult_minus_left
thf(fact_485_ereal__zero__times,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( times_7703590493115627913_ereal @ A @ B )
= zero_z2744965634713055877_ereal )
= ( ( A = zero_z2744965634713055877_ereal )
| ( B = zero_z2744965634713055877_ereal ) ) ) ).
% ereal_zero_times
thf(fact_486_ereal__zero__mult,axiom,
! [A: extended_ereal] :
( ( times_7703590493115627913_ereal @ zero_z2744965634713055877_ereal @ A )
= zero_z2744965634713055877_ereal ) ).
% ereal_zero_mult
thf(fact_487_ereal__mult__zero,axiom,
! [A: extended_ereal] :
( ( times_7703590493115627913_ereal @ A @ zero_z2744965634713055877_ereal )
= zero_z2744965634713055877_ereal ) ).
% ereal_mult_zero
thf(fact_488_ereal__mult__eq__PInfty,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( times_7703590493115627913_ereal @ A @ B )
= extend1530274965995635425_ereal )
= ( ( ( A = extend1530274965995635425_ereal )
& ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ B ) )
| ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ A )
& ( B = extend1530274965995635425_ereal ) )
| ( ( A
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
& ( ord_le1188267648640031866_ereal @ B @ zero_z2744965634713055877_ereal ) )
| ( ( ord_le1188267648640031866_ereal @ A @ zero_z2744965634713055877_ereal )
& ( B
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ).
% ereal_mult_eq_PInfty
thf(fact_489_ereal__mult__eq__MInfty,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( times_7703590493115627913_ereal @ A @ B )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
= ( ( ( A = extend1530274965995635425_ereal )
& ( ord_le1188267648640031866_ereal @ B @ zero_z2744965634713055877_ereal ) )
| ( ( ord_le1188267648640031866_ereal @ A @ zero_z2744965634713055877_ereal )
& ( B = extend1530274965995635425_ereal ) )
| ( ( A
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
& ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ B ) )
| ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ A )
& ( B
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ).
% ereal_mult_eq_MInfty
thf(fact_490_ereal__mult__infty,axiom,
! [A: extended_ereal] :
( ( ( A = zero_z2744965634713055877_ereal )
=> ( ( times_7703590493115627913_ereal @ A @ extend1530274965995635425_ereal )
= zero_z2744965634713055877_ereal ) )
& ( ( A != zero_z2744965634713055877_ereal )
=> ( ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( times_7703590493115627913_ereal @ A @ extend1530274965995635425_ereal )
= extend1530274965995635425_ereal ) )
& ( ~ ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( times_7703590493115627913_ereal @ A @ extend1530274965995635425_ereal )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ) ).
% ereal_mult_infty
thf(fact_491_ereal__infty__mult,axiom,
! [A: extended_ereal] :
( ( ( A = zero_z2744965634713055877_ereal )
=> ( ( times_7703590493115627913_ereal @ extend1530274965995635425_ereal @ A )
= zero_z2744965634713055877_ereal ) )
& ( ( A != zero_z2744965634713055877_ereal )
=> ( ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( times_7703590493115627913_ereal @ extend1530274965995635425_ereal @ A )
= extend1530274965995635425_ereal ) )
& ( ~ ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( times_7703590493115627913_ereal @ extend1530274965995635425_ereal @ A )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ) ).
% ereal_infty_mult
thf(fact_492_mult_Oleft__commute,axiom,
! [B: extended_ereal,A: extended_ereal,C2: extended_ereal] :
( ( times_7703590493115627913_ereal @ B @ ( times_7703590493115627913_ereal @ A @ C2 ) )
= ( times_7703590493115627913_ereal @ A @ ( times_7703590493115627913_ereal @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_493_mult_Oleft__commute,axiom,
! [B: real,A: real,C2: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C2 ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_494_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C2 ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_495_mult_Oleft__commute,axiom,
! [B: int,A: int,C2: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C2 ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_496_mult_Ocommute,axiom,
( times_7703590493115627913_ereal
= ( ^ [A2: extended_ereal,B2: extended_ereal] : ( times_7703590493115627913_ereal @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_497_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A2: real,B2: real] : ( times_times_real @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_498_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A2: nat,B2: nat] : ( times_times_nat @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_499_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A2: int,B2: int] : ( times_times_int @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_500_mult_Oassoc,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( times_7703590493115627913_ereal @ ( times_7703590493115627913_ereal @ A @ B ) @ C2 )
= ( times_7703590493115627913_ereal @ A @ ( times_7703590493115627913_ereal @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_501_mult_Oassoc,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C2 )
= ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_502_mult_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_503_mult_Oassoc,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C2 )
= ( times_times_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_504_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( times_7703590493115627913_ereal @ ( times_7703590493115627913_ereal @ A @ B ) @ C2 )
= ( times_7703590493115627913_ereal @ A @ ( times_7703590493115627913_ereal @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_505_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C2 )
= ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_506_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_507_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C2 )
= ( times_times_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_508_mult__not__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_509_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_510_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_511_divisors__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_512_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_513_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_514_no__zero__divisors,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_515_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_516_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_517_mult__left__cancel,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ C2 @ A )
= ( times_times_real @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_518_mult__left__cancel,axiom,
! [C2: nat,A: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_519_mult__left__cancel,axiom,
! [C2: int,A: int,B: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ C2 @ A )
= ( times_times_int @ C2 @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_520_mult__right__cancel,axiom,
! [C2: real,A: real,B: real] :
( ( C2 != zero_zero_real )
=> ( ( ( times_times_real @ A @ C2 )
= ( times_times_real @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_521_mult__right__cancel,axiom,
! [C2: nat,A: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_522_mult__right__cancel,axiom,
! [C2: int,A: int,B: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ A @ C2 )
= ( times_times_int @ B @ C2 ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_523_crossproduct__eq,axiom,
! [W: real,Y: real,X2: real,Z2: real] :
( ( ( plus_plus_real @ ( times_times_real @ W @ Y ) @ ( times_times_real @ X2 @ Z2 ) )
= ( plus_plus_real @ ( times_times_real @ W @ Z2 ) @ ( times_times_real @ X2 @ Y ) ) )
= ( ( W = X2 )
| ( Y = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_524_crossproduct__eq,axiom,
! [W: nat,Y: nat,X2: nat,Z2: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X2 @ Z2 ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z2 ) @ ( times_times_nat @ X2 @ Y ) ) )
= ( ( W = X2 )
| ( Y = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_525_crossproduct__eq,axiom,
! [W: int,Y: int,X2: int,Z2: int] :
( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X2 @ Z2 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z2 ) @ ( times_times_int @ X2 @ Y ) ) )
= ( ( W = X2 )
| ( Y = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_526_crossproduct__noteq,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ( A != B )
& ( C2 != D ) )
= ( ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) )
!= ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% crossproduct_noteq
thf(fact_527_crossproduct__noteq,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ( A != B )
& ( C2 != D ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).
% crossproduct_noteq
thf(fact_528_crossproduct__noteq,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ( A != B )
& ( C2 != D ) )
= ( ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) )
!= ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C2 ) ) ) ) ).
% crossproduct_noteq
thf(fact_529_ring__class_Oring__distribs_I2_J,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_530_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_531_ring__class_Oring__distribs_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_532_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_533_comm__semiring__class_Odistrib,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_534_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_535_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_536_distrib__left,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C2 ) ) ) ).
% distrib_left
thf(fact_537_distrib__left,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C2 ) ) ) ).
% distrib_left
thf(fact_538_distrib__left,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ).
% distrib_left
thf(fact_539_distrib__right,axiom,
! [A: real,B: real,C2: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).
% distrib_right
thf(fact_540_distrib__right,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ).
% distrib_right
thf(fact_541_distrib__right,axiom,
! [A: int,B: int,C2: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).
% distrib_right
thf(fact_542_combine__common__factor,axiom,
! [A: real,E2: real,B: real,C2: real] :
( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ C2 ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E2 ) @ C2 ) ) ).
% combine_common_factor
thf(fact_543_combine__common__factor,axiom,
! [A: nat,E2: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C2 ) ) ).
% combine_common_factor
thf(fact_544_combine__common__factor,axiom,
! [A: int,E2: int,B: int,C2: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E2 ) @ C2 ) ) ).
% combine_common_factor
thf(fact_545_times__ereal_Osimps_I6_J,axiom,
( ( times_7703590493115627913_ereal @ extend1530274965995635425_ereal @ extend1530274965995635425_ereal )
= extend1530274965995635425_ereal ) ).
% times_ereal.simps(6)
thf(fact_546_ereal__right__mult__cong,axiom,
! [C2: extended_ereal,D: extended_ereal,A: extended_ereal,B: extended_ereal] :
( ( C2 = D )
=> ( ( ( D != zero_z2744965634713055877_ereal )
=> ( A = B ) )
=> ( ( times_7703590493115627913_ereal @ C2 @ A )
= ( times_7703590493115627913_ereal @ D @ B ) ) ) ) ).
% ereal_right_mult_cong
thf(fact_547_ereal__left__mult__cong,axiom,
! [C2: extended_ereal,D: extended_ereal,A: extended_ereal,B: extended_ereal] :
( ( C2 = D )
=> ( ( ( D != zero_z2744965634713055877_ereal )
=> ( A = B ) )
=> ( ( times_7703590493115627913_ereal @ A @ C2 )
= ( times_7703590493115627913_ereal @ B @ D ) ) ) ) ).
% ereal_left_mult_cong
thf(fact_548_lambda__zero,axiom,
( ( ^ [H: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_549_lambda__zero,axiom,
( ( ^ [H: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_550_lambda__zero,axiom,
( ( ^ [H: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_551_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_552_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_553_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_554_mult__less__cancel__right__disj,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_555_mult__less__cancel__right__disj,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C2 @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_556_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_557_mult__strict__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_558_mult__strict__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_559_mult__strict__right__mono__neg,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_560_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_561_mult__less__cancel__left__disj,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C2 )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C2 @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_562_mult__less__cancel__left__disj,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C2 @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_563_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_564_mult__strict__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_565_mult__strict__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_566_mult__strict__left__mono__neg,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_567_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_568_mult__less__cancel__left__pos,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C2 )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ord_less_real @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_569_mult__less__cancel__left__pos,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_570_mult__less__cancel__left__neg,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ C2 @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
= ( ord_less_real @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_571_mult__less__cancel__left__neg,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_572_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_573_zero__less__mult__pos2,axiom,
! [B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_574_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_575_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_576_zero__less__mult__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_577_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_578_zero__less__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_579_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_580_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_581_mult__pos__neg2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_pos_neg2
thf(fact_582_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_583_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_584_mult__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_585_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_586_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_587_mult__pos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_pos_neg
thf(fact_588_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_589_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_590_mult__neg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_neg_pos
thf(fact_591_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_592_mult__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_593_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_594_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_595_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_596_mult__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_597_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_598_add__scale__eq__noteq,axiom,
! [R: real,A: real,B: real,C2: real,D: real] :
( ( R != zero_zero_real )
=> ( ( ( A = B )
& ( C2 != D ) )
=> ( ( plus_plus_real @ A @ ( times_times_real @ R @ C2 ) )
!= ( plus_plus_real @ B @ ( times_times_real @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_599_add__scale__eq__noteq,axiom,
! [R: nat,A: nat,B: nat,C2: nat,D: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A = B )
& ( C2 != D ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C2 ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_600_add__scale__eq__noteq,axiom,
! [R: int,A: int,B: int,C2: int,D: int] :
( ( R != zero_zero_int )
=> ( ( ( A = B )
& ( C2 != D ) )
=> ( ( plus_plus_int @ A @ ( times_times_int @ R @ C2 ) )
!= ( plus_plus_int @ B @ ( times_times_int @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_601_uminus__ereal_Osimps_I1_J,axiom,
! [R: real] :
( ( uminus27091377158695749_ereal @ ( extended_ereal2 @ R ) )
= ( extended_ereal2 @ ( uminus_uminus_real @ R ) ) ) ).
% uminus_ereal.simps(1)
thf(fact_602_times__ereal_Osimps_I7_J,axiom,
( ( times_7703590493115627913_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ extend1530274965995635425_ereal )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% times_ereal.simps(7)
thf(fact_603_times__ereal_Osimps_I8_J,axiom,
( ( times_7703590493115627913_ereal @ extend1530274965995635425_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% times_ereal.simps(8)
thf(fact_604_times__ereal_Osimps_I9_J,axiom,
( ( times_7703590493115627913_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
= extend1530274965995635425_ereal ) ).
% times_ereal.simps(9)
thf(fact_605_ereal__mult__less__0__iff,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ ( times_7703590493115627913_ereal @ A @ B ) @ zero_z2744965634713055877_ereal )
= ( ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ A )
& ( ord_le1188267648640031866_ereal @ B @ zero_z2744965634713055877_ereal ) )
| ( ( ord_le1188267648640031866_ereal @ A @ zero_z2744965634713055877_ereal )
& ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ B ) ) ) ) ).
% ereal_mult_less_0_iff
thf(fact_606_ereal__zero__less__0__iff,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ ( times_7703590493115627913_ereal @ A @ B ) )
= ( ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ A )
& ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ B ) )
| ( ( ord_le1188267648640031866_ereal @ A @ zero_z2744965634713055877_ereal )
& ( ord_le1188267648640031866_ereal @ B @ zero_z2744965634713055877_ereal ) ) ) ) ).
% ereal_zero_less_0_iff
thf(fact_607_ereal__mult__mono__strict,axiom,
! [B: extended_ereal,C2: extended_ereal,A: extended_ereal,D: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ B )
=> ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ C2 )
=> ( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ( ord_le1188267648640031866_ereal @ C2 @ D )
=> ( ord_le1188267648640031866_ereal @ ( times_7703590493115627913_ereal @ A @ C2 ) @ ( times_7703590493115627913_ereal @ B @ D ) ) ) ) ) ) ).
% ereal_mult_mono_strict
thf(fact_608_ereal__mult__mono__strict_H,axiom,
! [A: extended_ereal,C2: extended_ereal,B: extended_ereal,D: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ C2 )
=> ( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ( ord_le1188267648640031866_ereal @ C2 @ D )
=> ( ord_le1188267648640031866_ereal @ ( times_7703590493115627913_ereal @ A @ C2 ) @ ( times_7703590493115627913_ereal @ B @ D ) ) ) ) ) ) ).
% ereal_mult_mono_strict'
thf(fact_609_not__sum__squares__lt__zero,axiom,
! [X2: real,Y: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).
% not_sum_squares_lt_zero
thf(fact_610_not__sum__squares__lt__zero,axiom,
! [X2: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_611_real__0__less__add__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X2 @ Y ) )
= ( ord_less_real @ ( uminus_uminus_real @ X2 ) @ Y ) ) ).
% real_0_less_add_iff
thf(fact_612_real__add__less__0__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X2 @ Y ) @ zero_zero_real )
= ( ord_less_real @ Y @ ( uminus_uminus_real @ X2 ) ) ) ).
% real_add_less_0_iff
thf(fact_613_ereal__mult__strict__right__mono,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ C2 )
=> ( ( ord_le1188267648640031866_ereal @ C2 @ extend1530274965995635425_ereal )
=> ( ord_le1188267648640031866_ereal @ ( times_7703590493115627913_ereal @ A @ C2 ) @ ( times_7703590493115627913_ereal @ B @ C2 ) ) ) ) ) ).
% ereal_mult_strict_right_mono
thf(fact_614_ereal__mult__strict__left__mono,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ C2 )
=> ( ( ord_le1188267648640031866_ereal @ C2 @ extend1530274965995635425_ereal )
=> ( ord_le1188267648640031866_ereal @ ( times_7703590493115627913_ereal @ C2 @ A ) @ ( times_7703590493115627913_ereal @ C2 @ B ) ) ) ) ) ).
% ereal_mult_strict_left_mono
thf(fact_615_ereal__mult__less__right,axiom,
! [B: extended_ereal,A: extended_ereal,C2: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ ( times_7703590493115627913_ereal @ B @ A ) @ ( times_7703590493115627913_ereal @ C2 @ A ) )
=> ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( ord_le1188267648640031866_ereal @ A @ extend1530274965995635425_ereal )
=> ( ord_le1188267648640031866_ereal @ B @ C2 ) ) ) ) ).
% ereal_mult_less_right
thf(fact_616_uminus__ereal_Oelims,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( ( uminus27091377158695749_ereal @ X2 )
= Y )
=> ( ! [R2: real] :
( ( X2
= ( extended_ereal2 @ R2 ) )
=> ( Y
!= ( extended_ereal2 @ ( uminus_uminus_real @ R2 ) ) ) )
=> ( ( ( X2 = extended_PInfty )
=> ( Y != extended_MInfty ) )
=> ~ ( ( X2 = extended_MInfty )
=> ( Y != extended_PInfty ) ) ) ) ) ).
% uminus_ereal.elims
thf(fact_617_times__ereal_Osimps_I2_J,axiom,
! [R: real] :
( ( ( R = zero_zero_real )
=> ( ( times_7703590493115627913_ereal @ ( extended_ereal2 @ R ) @ extend1530274965995635425_ereal )
= zero_z2744965634713055877_ereal ) )
& ( ( R != zero_zero_real )
=> ( ( ( ord_less_real @ zero_zero_real @ R )
=> ( ( times_7703590493115627913_ereal @ ( extended_ereal2 @ R ) @ extend1530274965995635425_ereal )
= extend1530274965995635425_ereal ) )
& ( ~ ( ord_less_real @ zero_zero_real @ R )
=> ( ( times_7703590493115627913_ereal @ ( extended_ereal2 @ R ) @ extend1530274965995635425_ereal )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ) ).
% times_ereal.simps(2)
thf(fact_618_sum__squares__eq__zero__iff,axiom,
! [X2: real,Y: real] :
( ( ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y @ Y ) )
= zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_619_sum__squares__eq__zero__iff,axiom,
! [X2: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X2 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_620_sum__squares__gt__zero__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y @ Y ) ) )
= ( ( X2 != zero_zero_real )
| ( Y != zero_zero_real ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_621_sum__squares__gt__zero__iff,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X2 != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_622_mult__less__iff1,axiom,
! [Z2: real,X2: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_real @ ( times_times_real @ X2 @ Z2 ) @ ( times_times_real @ Y @ Z2 ) )
= ( ord_less_real @ X2 @ Y ) ) ) ).
% mult_less_iff1
thf(fact_623_mult__less__iff1,axiom,
! [Z2: int,X2: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_int @ ( times_times_int @ X2 @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
= ( ord_less_int @ X2 @ Y ) ) ) ).
% mult_less_iff1
thf(fact_624_times__ereal_Oelims,axiom,
! [X2: extended_ereal,Xa: extended_ereal,Y: extended_ereal] :
( ( ( times_7703590493115627913_ereal @ X2 @ Xa )
= Y )
=> ( ! [R2: real] :
( ( X2
= ( extended_ereal2 @ R2 ) )
=> ! [P3: real] :
( ( Xa
= ( extended_ereal2 @ P3 ) )
=> ( Y
!= ( extended_ereal2 @ ( times_times_real @ R2 @ P3 ) ) ) ) )
=> ( ! [R2: real] :
( ( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa = extend1530274965995635425_ereal )
=> ~ ( ( ( R2 = zero_zero_real )
=> ( Y = zero_z2744965634713055877_ereal ) )
& ( ( R2 != zero_zero_real )
=> ( ( ( ord_less_real @ zero_zero_real @ R2 )
=> ( Y = extend1530274965995635425_ereal ) )
& ( ~ ( ord_less_real @ zero_zero_real @ R2 )
=> ( Y
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ) ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ! [R2: real] :
( ( Xa
= ( extended_ereal2 @ R2 ) )
=> ~ ( ( ( R2 = zero_zero_real )
=> ( Y = zero_z2744965634713055877_ereal ) )
& ( ( R2 != zero_zero_real )
=> ( ( ( ord_less_real @ zero_zero_real @ R2 )
=> ( Y = extend1530274965995635425_ereal ) )
& ( ~ ( ord_less_real @ zero_zero_real @ R2 )
=> ( Y
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ) ) )
=> ( ! [R2: real] :
( ( X2
= ( extended_ereal2 @ R2 ) )
=> ( ( Xa
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ~ ( ( ( R2 = zero_zero_real )
=> ( Y = zero_z2744965634713055877_ereal ) )
& ( ( R2 != zero_zero_real )
=> ( ( ( ord_less_real @ zero_zero_real @ R2 )
=> ( Y
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ R2 )
=> ( Y = extend1530274965995635425_ereal ) ) ) ) ) ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ! [R2: real] :
( ( Xa
= ( extended_ereal2 @ R2 ) )
=> ~ ( ( ( R2 = zero_zero_real )
=> ( Y = zero_z2744965634713055877_ereal ) )
& ( ( R2 != zero_zero_real )
=> ( ( ( ord_less_real @ zero_zero_real @ R2 )
=> ( Y
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ R2 )
=> ( Y = extend1530274965995635425_ereal ) ) ) ) ) ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ( ( Xa = extend1530274965995635425_ereal )
=> ( Y != extend1530274965995635425_ereal ) ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( Xa = extend1530274965995635425_ereal )
=> ( Y
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ( ( Xa
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Y
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) )
=> ~ ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( Xa
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Y != extend1530274965995635425_ereal ) ) ) ) ) ) ) ) ) ) ) ) ).
% times_ereal.elims
thf(fact_625_zero__less__real__of__ereal,axiom,
! [X2: extended_ereal] :
( ( ord_less_real @ zero_zero_real @ ( extend2982805604970551563_ereal @ X2 ) )
= ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ X2 )
& ( X2 != extend1530274965995635425_ereal ) ) ) ).
% zero_less_real_of_ereal
thf(fact_626_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_627_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_628_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_629_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_630_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_631_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_632_real__of__ereal__0,axiom,
( ( extend2982805604970551563_ereal @ zero_z2744965634713055877_ereal )
= zero_zero_real ) ).
% real_of_ereal_0
thf(fact_633_real__of__ereal,axiom,
! [X2: extended_ereal] :
( ( extend2982805604970551563_ereal @ ( uminus27091377158695749_ereal @ X2 ) )
= ( uminus_uminus_real @ ( extend2982805604970551563_ereal @ X2 ) ) ) ).
% real_of_ereal
thf(fact_634_real__of__ereal__mult,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( extend2982805604970551563_ereal @ ( times_7703590493115627913_ereal @ A @ B ) )
= ( times_times_real @ ( extend2982805604970551563_ereal @ A ) @ ( extend2982805604970551563_ereal @ B ) ) ) ).
% real_of_ereal_mult
thf(fact_635_real__of__ereal_Osimps_I1_J,axiom,
! [R: real] :
( ( extend2982805604970551563_ereal @ ( extended_ereal2 @ R ) )
= R ) ).
% real_of_ereal.simps(1)
thf(fact_636_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_637_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_638_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_639_real__of__ereal_Osimps_I2_J,axiom,
( ( extend2982805604970551563_ereal @ extend1530274965995635425_ereal )
= zero_zero_real ) ).
% real_of_ereal.simps(2)
thf(fact_640_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_641_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_642_times__ereal_Osimps_I1_J,axiom,
! [R: real,P2: real] :
( ( times_7703590493115627913_ereal @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P2 ) )
= ( extended_ereal2 @ ( times_times_real @ R @ P2 ) ) ) ).
% times_ereal.simps(1)
thf(fact_643_real__of__ereal_Osimps_I3_J,axiom,
( ( extend2982805604970551563_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
= zero_zero_real ) ).
% real_of_ereal.simps(3)
thf(fact_644_real__of__ereal_Oelims,axiom,
! [X2: extended_ereal,Y: real] :
( ( ( extend2982805604970551563_ereal @ X2 )
= Y )
=> ( ! [R2: real] :
( ( X2
= ( extended_ereal2 @ R2 ) )
=> ( Y != R2 ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ( Y != zero_zero_real ) )
=> ~ ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Y != zero_zero_real ) ) ) ) ) ).
% real_of_ereal.elims
thf(fact_645_real__of__ereal__eq__0,axiom,
! [X2: extended_ereal] :
( ( ( extend2982805604970551563_ereal @ X2 )
= zero_zero_real )
= ( ( X2 = extend1530274965995635425_ereal )
| ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
| ( X2 = zero_z2744965634713055877_ereal ) ) ) ).
% real_of_ereal_eq_0
thf(fact_646_sum__list_Oeq__foldr,axiom,
( groups4561878855575611511st_nat
= ( ^ [Xs2: list_nat] : ( foldr_nat_nat @ plus_plus_nat @ Xs2 @ zero_zero_nat ) ) ) ).
% sum_list.eq_foldr
thf(fact_647_sum__list_Oeq__foldr,axiom,
( groups6723090944982001619t_real
= ( ^ [Xs2: list_real] : ( foldr_real_real @ plus_plus_real @ Xs2 @ zero_zero_real ) ) ) ).
% sum_list.eq_foldr
thf(fact_648_sum__list_Oeq__foldr,axiom,
( groups4559388385066561235st_int
= ( ^ [Xs2: list_int] : ( foldr_int_int @ plus_plus_int @ Xs2 @ zero_zero_int ) ) ) ).
% sum_list.eq_foldr
thf(fact_649_sum__list_Oeq__foldr,axiom,
( groups3567983573054521703_ereal
= ( ^ [Xs2: list_Extended_ereal] : ( foldr_6806735636647425191_ereal @ plus_p7876563987511257093_ereal @ Xs2 @ zero_z2744965634713055877_ereal ) ) ) ).
% sum_list.eq_foldr
thf(fact_650_not__real__square__gt__zero,axiom,
! [X2: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X2 @ X2 ) ) )
= ( X2 = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_651_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_652_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_653_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_654_real__less__ereal__iff,axiom,
! [Y: extended_ereal,X2: real] :
( ( ord_less_real @ ( extend2982805604970551563_ereal @ Y ) @ X2 )
= ( ( ( ( abs_ab7465543570706387889_ereal @ Y )
!= extend1530274965995635425_ereal )
=> ( ord_le1188267648640031866_ereal @ Y @ ( extended_ereal2 @ X2 ) ) )
& ( ( ( abs_ab7465543570706387889_ereal @ Y )
= extend1530274965995635425_ereal )
=> ( ord_less_real @ zero_zero_real @ X2 ) ) ) ) ).
% real_less_ereal_iff
thf(fact_655_abs__idempotent,axiom,
! [A: real] :
( ( abs_abs_real @ ( abs_abs_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_idempotent
thf(fact_656_abs__idempotent,axiom,
! [A: int] :
( ( abs_abs_int @ ( abs_abs_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_idempotent
thf(fact_657_abs__zero,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_zero
thf(fact_658_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_zero
thf(fact_659_abs__eq__0,axiom,
! [A: real] :
( ( ( abs_abs_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_eq_0
thf(fact_660_abs__eq__0,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0
thf(fact_661_abs__0__eq,axiom,
! [A: real] :
( ( zero_zero_real
= ( abs_abs_real @ A ) )
= ( A = zero_zero_real ) ) ).
% abs_0_eq
thf(fact_662_abs__0__eq,axiom,
! [A: int] :
( ( zero_zero_int
= ( abs_abs_int @ A ) )
= ( A = zero_zero_int ) ) ).
% abs_0_eq
thf(fact_663_abs__0,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_0
thf(fact_664_abs__0,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_0
thf(fact_665_abs__add__abs,axiom,
! [A: real,B: real] :
( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
= ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% abs_add_abs
thf(fact_666_abs__add__abs,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
= ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_add_abs
thf(fact_667_abs__minus__cancel,axiom,
! [A: real] :
( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_minus_cancel
thf(fact_668_abs__minus__cancel,axiom,
! [A: int] :
( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_minus_cancel
thf(fact_669_abs__ereal__uminus,axiom,
! [X2: extended_ereal] :
( ( abs_ab7465543570706387889_ereal @ ( uminus27091377158695749_ereal @ X2 ) )
= ( abs_ab7465543570706387889_ereal @ X2 ) ) ).
% abs_ereal_uminus
thf(fact_670_abs__ereal__zero,axiom,
( ( abs_ab7465543570706387889_ereal @ zero_z2744965634713055877_ereal )
= zero_z2744965634713055877_ereal ) ).
% abs_ereal_zero
thf(fact_671_zero__less__abs__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
= ( A != zero_zero_real ) ) ).
% zero_less_abs_iff
thf(fact_672_zero__less__abs__iff,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
= ( A != zero_zero_int ) ) ).
% zero_less_abs_iff
thf(fact_673_abs__ereal__less0,axiom,
! [X2: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X2 @ zero_z2744965634713055877_ereal )
=> ( ( abs_ab7465543570706387889_ereal @ X2 )
= ( uminus27091377158695749_ereal @ X2 ) ) ) ).
% abs_ereal_less0
thf(fact_674_abs__ereal_Osimps_I3_J,axiom,
( ( abs_ab7465543570706387889_ereal @ extend1530274965995635425_ereal )
= extend1530274965995635425_ereal ) ).
% abs_ereal.simps(3)
thf(fact_675_ereal__abs__mult,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( abs_ab7465543570706387889_ereal @ ( times_7703590493115627913_ereal @ X2 @ Y ) )
= ( times_7703590493115627913_ereal @ ( abs_ab7465543570706387889_ereal @ X2 ) @ ( abs_ab7465543570706387889_ereal @ Y ) ) ) ).
% ereal_abs_mult
thf(fact_676_abs__eq__0__iff,axiom,
! [A: real] :
( ( ( abs_abs_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_eq_0_iff
thf(fact_677_abs__eq__0__iff,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0_iff
thf(fact_678_abs__not__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% abs_not_less_zero
thf(fact_679_abs__not__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% abs_not_less_zero
thf(fact_680_abs__of__pos,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( abs_abs_real @ A )
= A ) ) ).
% abs_of_pos
thf(fact_681_abs__of__pos,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_pos
thf(fact_682_abs__mult__less,axiom,
! [A: real,C2: real,B: real,D: real] :
( ( ord_less_real @ ( abs_abs_real @ A ) @ C2 )
=> ( ( ord_less_real @ ( abs_abs_real @ B ) @ D )
=> ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C2 @ D ) ) ) ) ).
% abs_mult_less
thf(fact_683_abs__mult__less,axiom,
! [A: int,C2: int,B: int,D: int] :
( ( ord_less_int @ ( abs_abs_int @ A ) @ C2 )
=> ( ( ord_less_int @ ( abs_abs_int @ B ) @ D )
=> ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C2 @ D ) ) ) ) ).
% abs_mult_less
thf(fact_684_abs__less__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
= ( ( ord_less_real @ A @ B )
& ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% abs_less_iff
thf(fact_685_abs__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
= ( ( ord_less_int @ A @ B )
& ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% abs_less_iff
thf(fact_686_not__infty__ereal,axiom,
! [X2: extended_ereal] :
( ( ( abs_ab7465543570706387889_ereal @ X2 )
!= extend1530274965995635425_ereal )
= ( ? [X10: real] :
( X2
= ( extended_ereal2 @ X10 ) ) ) ) ).
% not_infty_ereal
thf(fact_687_abs__neq__infinity__cases,axiom,
! [X2: extended_ereal] :
( ( ( abs_ab7465543570706387889_ereal @ X2 )
!= extend1530274965995635425_ereal )
=> ~ ! [R2: real] :
( X2
!= ( extended_ereal2 @ R2 ) ) ) ).
% abs_neq_infinity_cases
thf(fact_688_abs__eq__infinity__cases,axiom,
! [X2: extended_ereal] :
( ( ( abs_ab7465543570706387889_ereal @ X2 )
= extend1530274965995635425_ereal )
=> ( ( X2 != extend1530274965995635425_ereal )
=> ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ).
% abs_eq_infinity_cases
thf(fact_689_ereal__infinity__cases,axiom,
! [A: extended_ereal] :
( ( A != extend1530274965995635425_ereal )
=> ( ( A
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( abs_ab7465543570706387889_ereal @ A )
!= extend1530274965995635425_ereal ) ) ) ).
% ereal_infinity_cases
thf(fact_690_abs__ereal_Osimps_I2_J,axiom,
( ( abs_ab7465543570706387889_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
= extend1530274965995635425_ereal ) ).
% abs_ereal.simps(2)
thf(fact_691_not__inftyI,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ( ord_le1188267648640031866_ereal @ B @ C2 )
=> ( ( abs_ab7465543570706387889_ereal @ B )
!= extend1530274965995635425_ereal ) ) ) ).
% not_inftyI
thf(fact_692_abs__if__raw,axiom,
( abs_abs_real
= ( ^ [A2: real] : ( if_real @ ( ord_less_real @ A2 @ zero_zero_real ) @ ( uminus_uminus_real @ A2 ) @ A2 ) ) ) ).
% abs_if_raw
thf(fact_693_abs__if__raw,axiom,
( abs_abs_int
= ( ^ [A2: int] : ( if_int @ ( ord_less_int @ A2 @ zero_zero_int ) @ ( uminus_uminus_int @ A2 ) @ A2 ) ) ) ).
% abs_if_raw
thf(fact_694_abs__if__lattice,axiom,
( abs_abs_real
= ( ^ [A2: real] : ( if_real @ ( ord_less_real @ A2 @ zero_zero_real ) @ ( uminus_uminus_real @ A2 ) @ A2 ) ) ) ).
% abs_if_lattice
thf(fact_695_abs__if__lattice,axiom,
( abs_abs_int
= ( ^ [A2: int] : ( if_int @ ( ord_less_int @ A2 @ zero_zero_int ) @ ( uminus_uminus_int @ A2 ) @ A2 ) ) ) ).
% abs_if_lattice
thf(fact_696_abs__of__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( abs_abs_real @ A )
= ( uminus_uminus_real @ A ) ) ) ).
% abs_of_neg
thf(fact_697_abs__of__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( abs_abs_int @ A )
= ( uminus_uminus_int @ A ) ) ) ).
% abs_of_neg
thf(fact_698_abs__if,axiom,
( abs_abs_real
= ( ^ [A2: real] : ( if_real @ ( ord_less_real @ A2 @ zero_zero_real ) @ ( uminus_uminus_real @ A2 ) @ A2 ) ) ) ).
% abs_if
thf(fact_699_abs__if,axiom,
( abs_abs_int
= ( ^ [A2: int] : ( if_int @ ( ord_less_int @ A2 @ zero_zero_int ) @ ( uminus_uminus_int @ A2 ) @ A2 ) ) ) ).
% abs_if
thf(fact_700_ereal__real_H,axiom,
! [X2: extended_ereal] :
( ( ( abs_ab7465543570706387889_ereal @ X2 )
!= extend1530274965995635425_ereal )
=> ( ( extended_ereal2 @ ( extend2982805604970551563_ereal @ X2 ) )
= X2 ) ) ).
% ereal_real'
thf(fact_701_ereal__less__add,axiom,
! [A: extended_ereal,C2: extended_ereal,B: extended_ereal] :
( ( ( abs_ab7465543570706387889_ereal @ A )
!= extend1530274965995635425_ereal )
=> ( ( ord_le1188267648640031866_ereal @ C2 @ B )
=> ( ord_le1188267648640031866_ereal @ ( plus_p7876563987511257093_ereal @ A @ C2 ) @ ( plus_p7876563987511257093_ereal @ A @ B ) ) ) ) ).
% ereal_less_add
thf(fact_702_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_703_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_704_ereal__distrib,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( ( A != extend1530274965995635425_ereal )
| ( B
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) )
=> ( ( ( A
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
| ( B != extend1530274965995635425_ereal ) )
=> ( ( ( abs_ab7465543570706387889_ereal @ C2 )
!= extend1530274965995635425_ereal )
=> ( ( times_7703590493115627913_ereal @ ( plus_p7876563987511257093_ereal @ A @ B ) @ C2 )
= ( plus_p7876563987511257093_ereal @ ( times_7703590493115627913_ereal @ A @ C2 ) @ ( times_7703590493115627913_ereal @ B @ C2 ) ) ) ) ) ) ).
% ereal_distrib
thf(fact_705_ereal__distrib__left,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( ( A != extend1530274965995635425_ereal )
| ( B
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) )
=> ( ( ( A
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
| ( B != extend1530274965995635425_ereal ) )
=> ( ( ( abs_ab7465543570706387889_ereal @ C2 )
!= extend1530274965995635425_ereal )
=> ( ( times_7703590493115627913_ereal @ C2 @ ( plus_p7876563987511257093_ereal @ A @ B ) )
= ( plus_p7876563987511257093_ereal @ ( times_7703590493115627913_ereal @ C2 @ A ) @ ( times_7703590493115627913_ereal @ C2 @ B ) ) ) ) ) ) ).
% ereal_distrib_left
thf(fact_706_ereal__real,axiom,
! [X2: extended_ereal] :
( ( ( ( abs_ab7465543570706387889_ereal @ X2 )
= extend1530274965995635425_ereal )
=> ( ( extended_ereal2 @ ( extend2982805604970551563_ereal @ X2 ) )
= zero_z2744965634713055877_ereal ) )
& ( ( ( abs_ab7465543570706387889_ereal @ X2 )
!= extend1530274965995635425_ereal )
=> ( ( extended_ereal2 @ ( extend2982805604970551563_ereal @ X2 ) )
= X2 ) ) ) ).
% ereal_real
thf(fact_707_ereal__mult__cancel__left,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( ( times_7703590493115627913_ereal @ A @ B )
= ( times_7703590493115627913_ereal @ A @ C2 ) )
= ( ( ( ( abs_ab7465543570706387889_ereal @ A )
= extend1530274965995635425_ereal )
& ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ ( times_7703590493115627913_ereal @ B @ C2 ) ) )
| ( A = zero_z2744965634713055877_ereal )
| ( B = C2 ) ) ) ).
% ereal_mult_cancel_left
thf(fact_708_ereal__between_I2_J,axiom,
! [X2: extended_ereal,E2: extended_ereal] :
( ( ( abs_ab7465543570706387889_ereal @ X2 )
!= extend1530274965995635425_ereal )
=> ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ E2 )
=> ( ord_le1188267648640031866_ereal @ X2 @ ( plus_p7876563987511257093_ereal @ X2 @ E2 ) ) ) ) ).
% ereal_between(2)
thf(fact_709_real__of__ereal__add,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( ( ( ( abs_ab7465543570706387889_ereal @ A )
= extend1530274965995635425_ereal )
& ( ( abs_ab7465543570706387889_ereal @ B )
= extend1530274965995635425_ereal ) )
| ( ( ( abs_ab7465543570706387889_ereal @ A )
!= extend1530274965995635425_ereal )
& ( ( abs_ab7465543570706387889_ereal @ B )
!= extend1530274965995635425_ereal ) ) )
=> ( ( extend2982805604970551563_ereal @ ( plus_p7876563987511257093_ereal @ A @ B ) )
= ( plus_plus_real @ ( extend2982805604970551563_ereal @ A ) @ ( extend2982805604970551563_ereal @ B ) ) ) )
& ( ~ ( ( ( ( abs_ab7465543570706387889_ereal @ A )
= extend1530274965995635425_ereal )
& ( ( abs_ab7465543570706387889_ereal @ B )
= extend1530274965995635425_ereal ) )
| ( ( ( abs_ab7465543570706387889_ereal @ A )
!= extend1530274965995635425_ereal )
& ( ( abs_ab7465543570706387889_ereal @ B )
!= extend1530274965995635425_ereal ) ) )
=> ( ( extend2982805604970551563_ereal @ ( plus_p7876563987511257093_ereal @ A @ B ) )
= zero_zero_real ) ) ) ).
% real_of_ereal_add
thf(fact_710_ereal__less__real__iff,axiom,
! [X2: real,Y: extended_ereal] :
( ( ord_less_real @ X2 @ ( extend2982805604970551563_ereal @ Y ) )
= ( ( ( ( abs_ab7465543570706387889_ereal @ Y )
!= extend1530274965995635425_ereal )
=> ( ord_le1188267648640031866_ereal @ ( extended_ereal2 @ X2 ) @ Y ) )
& ( ( ( abs_ab7465543570706387889_ereal @ Y )
= extend1530274965995635425_ereal )
=> ( ord_less_real @ X2 @ zero_zero_real ) ) ) ) ).
% ereal_less_real_iff
thf(fact_711_ereal__mult__m1,axiom,
! [X2: extended_ereal] :
( ( times_7703590493115627913_ereal @ X2 @ ( extended_ereal2 @ ( uminus_uminus_real @ one_one_real ) ) )
= ( uminus27091377158695749_ereal @ X2 ) ) ).
% ereal_mult_m1
thf(fact_712_length__product,axiom,
! [Xs: list_a,Ys: list_a] :
( ( size_s3885678630836030617od_a_a @ ( product_a_a @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) ) ) ).
% length_product
thf(fact_713_length__product,axiom,
! [Xs: list_a,Ys: list_Extended_ereal] :
( ( size_s6930574783941632489_ereal @ ( produc2373813038293282581_ereal @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_a @ Xs ) @ ( size_s2768339837476157504_ereal @ Ys ) ) ) ).
% length_product
thf(fact_714_length__product,axiom,
! [Xs: list_Extended_ereal,Ys: list_a] :
( ( size_s6587793725349607369real_a @ ( produc6916318822199699189real_a @ Xs @ Ys ) )
= ( times_times_nat @ ( size_s2768339837476157504_ereal @ Xs ) @ ( size_size_list_a @ Ys ) ) ) ).
% length_product
thf(fact_715_length__product,axiom,
! [Xs: list_Extended_ereal,Ys: list_Extended_ereal] :
( ( size_s2383823777779303705_ereal @ ( produc2704682175734110533_ereal @ Xs @ Ys ) )
= ( times_times_nat @ ( size_s2768339837476157504_ereal @ Xs ) @ ( size_s2768339837476157504_ereal @ Ys ) ) ) ).
% length_product
thf(fact_716_ereal__mult__le__mult__iff,axiom,
! [C2: extended_ereal,A: extended_ereal,B: extended_ereal] :
( ( ( abs_ab7465543570706387889_ereal @ C2 )
!= extend1530274965995635425_ereal )
=> ( ( ord_le1083603963089353582_ereal @ ( times_7703590493115627913_ereal @ C2 @ A ) @ ( times_7703590493115627913_ereal @ C2 @ B ) )
= ( ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ C2 )
=> ( ord_le1083603963089353582_ereal @ A @ B ) )
& ( ( ord_le1188267648640031866_ereal @ C2 @ zero_z2744965634713055877_ereal )
=> ( ord_le1083603963089353582_ereal @ B @ A ) ) ) ) ) ).
% ereal_mult_le_mult_iff
thf(fact_717_ereal__less__minus__iff,axiom,
! [X2: extended_ereal,Z2: extended_ereal,Y: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X2 @ ( minus_2816186181549245109_ereal @ Z2 @ Y ) )
= ( ( ( Y = extend1530274965995635425_ereal )
=> ( ( Z2 = extend1530274965995635425_ereal )
& ( X2 != extend1530274965995635425_ereal ) ) )
& ( ( Y
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( X2 != extend1530274965995635425_ereal ) )
& ( ( ( abs_ab7465543570706387889_ereal @ Y )
!= extend1530274965995635425_ereal )
=> ( ord_le1188267648640031866_ereal @ ( plus_p7876563987511257093_ereal @ X2 @ Y ) @ Z2 ) ) ) ) ).
% ereal_less_minus_iff
thf(fact_718_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_719_add__le__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_720_add__le__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_721_add__le__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_722_add__le__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_723_add__le__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_724_add__le__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_725_neg__le__iff__le,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_726_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_727_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_728_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_729_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_730_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_731_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_732_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_733_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_734_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_735_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_736_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_737_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_738_mult_Oright__neutral,axiom,
! [A: extended_ereal] :
( ( times_7703590493115627913_ereal @ A @ one_on4623092294121504201_ereal )
= A ) ).
% mult.right_neutral
thf(fact_739_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_740_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_741_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_742_mult__1,axiom,
! [A: extended_ereal] :
( ( times_7703590493115627913_ereal @ one_on4623092294121504201_ereal @ A )
= A ) ).
% mult_1
thf(fact_743_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_744_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_745_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_746_add__diff__cancel,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_747_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_748_diff__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_749_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_750_add__diff__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_751_add__diff__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_752_add__diff__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_753_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_754_add__diff__cancel__left_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_755_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_756_add__diff__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_757_add__diff__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_758_add__diff__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_759_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_760_add__diff__cancel__right_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_761_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_762_minus__diff__eq,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
= ( minus_minus_real @ B @ A ) ) ).
% minus_diff_eq
thf(fact_763_minus__diff__eq,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ B @ A ) ) ).
% minus_diff_eq
thf(fact_764_ereal__infty__less__eq_I1_J,axiom,
! [X2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ extend1530274965995635425_ereal @ X2 )
= ( X2 = extend1530274965995635425_ereal ) ) ).
% ereal_infty_less_eq(1)
thf(fact_765_ereal__minus__le__minus,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ A ) @ ( uminus27091377158695749_ereal @ B ) )
= ( ord_le1083603963089353582_ereal @ B @ A ) ) ).
% ereal_minus_le_minus
thf(fact_766_ereal__minus_I4_J,axiom,
! [X2: extended_ereal] :
( ( minus_2816186181549245109_ereal @ extend1530274965995635425_ereal @ X2 )
= extend1530274965995635425_ereal ) ).
% ereal_minus(4)
thf(fact_767_ereal__minus_I7_J,axiom,
! [X2: extended_ereal] :
( ( minus_2816186181549245109_ereal @ X2 @ zero_z2744965634713055877_ereal )
= X2 ) ).
% ereal_minus(7)
thf(fact_768_abs__real__of__ereal,axiom,
! [X2: extended_ereal] :
( ( abs_abs_real @ ( extend2982805604970551563_ereal @ X2 ) )
= ( extend2982805604970551563_ereal @ ( abs_ab7465543570706387889_ereal @ X2 ) ) ) ).
% abs_real_of_ereal
thf(fact_769_lattice__ab__group__add__class_Ozero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% lattice_ab_group_add_class.zero_le_double_add_iff_zero_le_single_add
thf(fact_770_lattice__ab__group__add__class_Ozero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% lattice_ab_group_add_class.zero_le_double_add_iff_zero_le_single_add
thf(fact_771_lattice__ab__group__add__class_Odouble__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% lattice_ab_group_add_class.double_add_le_zero_iff_single_add_le_zero
thf(fact_772_lattice__ab__group__add__class_Odouble__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% lattice_ab_group_add_class.double_add_le_zero_iff_single_add_le_zero
thf(fact_773_linordered__ab__group__add__class_Ozero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% linordered_ab_group_add_class.zero_le_double_add_iff_zero_le_single_add
thf(fact_774_linordered__ab__group__add__class_Ozero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% linordered_ab_group_add_class.zero_le_double_add_iff_zero_le_single_add
thf(fact_775_linordered__ab__group__add__class_Odouble__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% linordered_ab_group_add_class.double_add_le_zero_iff_single_add_le_zero
thf(fact_776_linordered__ab__group__add__class_Odouble__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% linordered_ab_group_add_class.double_add_le_zero_iff_single_add_le_zero
thf(fact_777_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_778_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_779_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_780_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_781_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_782_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_783_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_784_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_785_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_786_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_787_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_788_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_789_neg__0__le__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_790_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_791_neg__le__0__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_le_0_iff_le
thf(fact_792_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_793_less__eq__neg__nonpos,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_794_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_795_neg__less__eq__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_796_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_797_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_798_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_799_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_800_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_801_mult__cancel__right2,axiom,
! [A: real,C2: real] :
( ( ( times_times_real @ A @ C2 )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_802_mult__cancel__right2,axiom,
! [A: int,C2: int] :
( ( ( times_times_int @ A @ C2 )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_803_mult__cancel__right1,axiom,
! [C2: real,B: real] :
( ( C2
= ( times_times_real @ B @ C2 ) )
= ( ( C2 = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_804_mult__cancel__right1,axiom,
! [C2: int,B: int] :
( ( C2
= ( times_times_int @ B @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_805_mult__cancel__left2,axiom,
! [C2: real,A: real] :
( ( ( times_times_real @ C2 @ A )
= C2 )
= ( ( C2 = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_806_mult__cancel__left2,axiom,
! [C2: int,A: int] :
( ( ( times_times_int @ C2 @ A )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_807_mult__cancel__left1,axiom,
! [C2: real,B: real] :
( ( C2
= ( times_times_real @ C2 @ B ) )
= ( ( C2 = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_808_mult__cancel__left1,axiom,
! [C2: int,B: int] :
( ( C2
= ( times_times_int @ C2 @ B ) )
= ( ( C2 = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_809_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_810_le__add__diff__inverse2,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_811_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_812_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_813_le__add__diff__inverse,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_814_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_815_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_816_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_817_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_818_diff__0,axiom,
! [A: real] :
( ( minus_minus_real @ zero_zero_real @ A )
= ( uminus_uminus_real @ A ) ) ).
% diff_0
thf(fact_819_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_820_verit__minus__simplify_I3_J,axiom,
! [B: real] :
( ( minus_minus_real @ zero_zero_real @ B )
= ( uminus_uminus_real @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_821_verit__minus__simplify_I3_J,axiom,
! [B: int] :
( ( minus_minus_int @ zero_zero_int @ B )
= ( uminus_uminus_int @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_822_abs__of__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( abs_abs_real @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_823_abs__of__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_824_abs__le__self__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% abs_le_self_iff
thf(fact_825_abs__le__self__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% abs_le_self_iff
thf(fact_826_abs__le__zero__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_le_zero_iff
thf(fact_827_abs__le__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_le_zero_iff
thf(fact_828_diff__minus__eq__add,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
= ( plus_plus_real @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_829_diff__minus__eq__add,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_830_uminus__add__conv__diff,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
= ( minus_minus_real @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_831_uminus__add__conv__diff,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_832_ereal__infty__less__eq_I2_J,axiom,
! [X2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X2 @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
= ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ).
% ereal_infty_less_eq(2)
thf(fact_833_ereal__1__times,axiom,
! [X2: extended_ereal] :
( ( times_7703590493115627913_ereal @ ( extended_ereal2 @ one_one_real ) @ X2 )
= X2 ) ).
% ereal_1_times
thf(fact_834_times__ereal__1,axiom,
! [X2: extended_ereal] :
( ( times_7703590493115627913_ereal @ X2 @ ( extended_ereal2 @ one_one_real ) )
= X2 ) ).
% times_ereal_1
thf(fact_835_ereal__uminus__le__0__iff,axiom,
! [A: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ A ) @ zero_z2744965634713055877_ereal )
= ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A ) ) ).
% ereal_uminus_le_0_iff
thf(fact_836_ereal__0__le__uminus__iff,axiom,
! [A: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( uminus27091377158695749_ereal @ A ) )
= ( ord_le1083603963089353582_ereal @ A @ zero_z2744965634713055877_ereal ) ) ).
% ereal_0_le_uminus_iff
thf(fact_837_minus__ereal__0,axiom,
! [X2: extended_ereal] :
( ( minus_2816186181549245109_ereal @ X2 @ ( extended_ereal2 @ zero_zero_real ) )
= X2 ) ).
% minus_ereal_0
thf(fact_838_ereal__minus_I5_J,axiom,
( ( minus_2816186181549245109_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ extend1530274965995635425_ereal )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% ereal_minus(5)
thf(fact_839_abs__ereal__ge0,axiom,
! [X2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ X2 )
=> ( ( abs_ab7465543570706387889_ereal @ X2 )
= X2 ) ) ).
% abs_ereal_ge0
thf(fact_840_ereal__minus_I8_J,axiom,
! [X2: extended_ereal] :
( ( minus_2816186181549245109_ereal @ zero_z2744965634713055877_ereal @ X2 )
= ( uminus27091377158695749_ereal @ X2 ) ) ).
% ereal_minus(8)
thf(fact_841_ereal__minus_I6_J,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( minus_2816186181549245109_ereal @ X2 @ ( uminus27091377158695749_ereal @ Y ) )
= ( plus_p7876563987511257093_ereal @ X2 @ Y ) ) ).
% ereal_minus(6)
thf(fact_842_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% add_neg_numeral_special(7)
thf(fact_843_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_844_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
= zero_zero_real ) ).
% add_neg_numeral_special(8)
thf(fact_845_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_846_diff__numeral__special_I12_J,axiom,
( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% diff_numeral_special(12)
thf(fact_847_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_848_abs__of__nonpos,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( abs_abs_real @ A )
= ( uminus_uminus_real @ A ) ) ) ).
% abs_of_nonpos
thf(fact_849_abs__of__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( abs_abs_int @ A )
= ( uminus_uminus_int @ A ) ) ) ).
% abs_of_nonpos
thf(fact_850_ereal__minus_I2_J,axiom,
! [R: real] :
( ( minus_2816186181549245109_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ ( extended_ereal2 @ R ) )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% ereal_minus(2)
thf(fact_851_ereal__minus_I3_J,axiom,
! [R: real] :
( ( minus_2816186181549245109_ereal @ ( extended_ereal2 @ R ) @ extend1530274965995635425_ereal )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% ereal_minus(3)
thf(fact_852_ereal__x__minus__x,axiom,
! [X2: extended_ereal] :
( ( ( ( abs_ab7465543570706387889_ereal @ X2 )
= extend1530274965995635425_ereal )
=> ( ( minus_2816186181549245109_ereal @ X2 @ X2 )
= extend1530274965995635425_ereal ) )
& ( ( ( abs_ab7465543570706387889_ereal @ X2 )
!= extend1530274965995635425_ereal )
=> ( ( minus_2816186181549245109_ereal @ X2 @ X2 )
= zero_z2744965634713055877_ereal ) ) ) ).
% ereal_x_minus_x
thf(fact_853_ordered__ring__class_Ole__add__iff1,axiom,
! [A: real,E2: real,C2: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
= ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C2 ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_854_ordered__ring__class_Ole__add__iff1,axiom,
! [A: int,E2: int,C2: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C2 ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_855_ordered__ring__class_Ole__add__iff2,axiom,
! [A: real,E2: real,C2: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
= ( ord_less_eq_real @ C2 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_856_ordered__ring__class_Ole__add__iff2,axiom,
! [A: int,E2: int,C2: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ord_less_eq_int @ C2 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_857_square__diff__one__factored,axiom,
! [X2: real] :
( ( minus_minus_real @ ( times_times_real @ X2 @ X2 ) @ one_one_real )
= ( times_times_real @ ( plus_plus_real @ X2 @ one_one_real ) @ ( minus_minus_real @ X2 @ one_one_real ) ) ) ).
% square_diff_one_factored
thf(fact_858_square__diff__one__factored,axiom,
! [X2: int] :
( ( minus_minus_int @ ( times_times_int @ X2 @ X2 ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X2 @ one_one_int ) @ ( minus_minus_int @ X2 @ one_one_int ) ) ) ).
% square_diff_one_factored
thf(fact_859_mult__left__le__one__le,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y @ X2 ) @ X2 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_860_mult__left__le__one__le,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X2 ) @ X2 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_861_mult__right__le__one__le,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X2 @ Y ) @ X2 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_862_mult__right__le__one__le,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X2 @ Y ) @ X2 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_863_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_864_mult__le__one,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_865_mult__le__one,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_866_mult__left__le,axiom,
! [C2: nat,A: nat] :
( ( ord_less_eq_nat @ C2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ A ) ) ) ).
% mult_left_le
thf(fact_867_mult__left__le,axiom,
! [C2: real,A: real] :
( ( ord_less_eq_real @ C2 @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ A ) ) ) ).
% mult_left_le
thf(fact_868_mult__left__le,axiom,
! [C2: int,A: int] :
( ( ord_less_eq_int @ C2 @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ A ) ) ) ).
% mult_left_le
thf(fact_869_comm__monoid__mult__class_Omult__1,axiom,
! [A: extended_ereal] :
( ( times_7703590493115627913_ereal @ one_on4623092294121504201_ereal @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_870_comm__monoid__mult__class_Omult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_871_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_872_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_873_mult_Ocomm__neutral,axiom,
! [A: extended_ereal] :
( ( times_7703590493115627913_ereal @ A @ one_on4623092294121504201_ereal )
= A ) ).
% mult.comm_neutral
thf(fact_874_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_875_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_876_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_877_abs__diff__le__iff,axiom,
! [X2: real,A: real,R: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ A ) ) @ R )
= ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R ) @ X2 )
& ( ord_less_eq_real @ X2 @ ( plus_plus_real @ A @ R ) ) ) ) ).
% abs_diff_le_iff
thf(fact_878_abs__diff__le__iff,axiom,
! [X2: int,A: int,R: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ A ) ) @ R )
= ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R ) @ X2 )
& ( ord_less_eq_int @ X2 @ ( plus_plus_int @ A @ R ) ) ) ) ).
% abs_diff_le_iff
thf(fact_879_abs__triangle__ineq2__sym,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_880_abs__triangle__ineq2__sym,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_881_abs__triangle__ineq3,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% abs_triangle_ineq3
thf(fact_882_abs__triangle__ineq3,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% abs_triangle_ineq3
thf(fact_883_abs__triangle__ineq2,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% abs_triangle_ineq2
thf(fact_884_abs__triangle__ineq2,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% abs_triangle_ineq2
thf(fact_885_abs__minus__commute,axiom,
! [A: real,B: real] :
( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
= ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_886_abs__minus__commute,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
= ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_887_abs__ge__self,axiom,
! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% abs_ge_self
thf(fact_888_abs__ge__self,axiom,
! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% abs_ge_self
thf(fact_889_abs__le__D1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% abs_le_D1
thf(fact_890_abs__le__D1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% abs_le_D1
thf(fact_891_abs__triangle__ineq4,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% abs_triangle_ineq4
thf(fact_892_abs__triangle__ineq4,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_triangle_ineq4
thf(fact_893_abs__diff__triangle__ineq,axiom,
! [A: real,B: real,C2: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C2 @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C2 ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_894_abs__diff__triangle__ineq,axiom,
! [A: int,B: int,C2: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C2 @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_895_ereal__abs__diff,axiom,
! [A: extended_ereal,B: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( abs_ab7465543570706387889_ereal @ ( minus_2816186181549245109_ereal @ A @ B ) ) @ ( plus_p7876563987511257093_ereal @ ( abs_ab7465543570706387889_ereal @ A ) @ ( abs_ab7465543570706387889_ereal @ B ) ) ) ).
% ereal_abs_diff
thf(fact_896_less__eq__ereal__def,axiom,
( ord_le1083603963089353582_ereal
= ( ^ [X: extended_ereal,Y9: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X @ Y9 )
| ( X = Y9 ) ) ) ) ).
% less_eq_ereal_def
thf(fact_897_diff__le__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C2 )
= ( ord_less_eq_real @ A @ ( plus_plus_real @ C2 @ B ) ) ) ).
% diff_le_eq
thf(fact_898_diff__le__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_le_eq
thf(fact_899_le__diff__eq,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ A @ ( minus_minus_real @ C2 @ B ) )
= ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C2 ) ) ).
% le_diff_eq
thf(fact_900_le__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C2 @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% le_diff_eq
thf(fact_901_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_902_le__add__diff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).
% le_add_diff
thf(fact_903_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_904_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_905_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A )
= ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_906_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_907_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_908_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_909_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_910_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C2 )
= ( B
= ( plus_plus_nat @ C2 @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_911_diff__eq__diff__eq,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C2 @ D ) )
=> ( ( A = B )
= ( C2 = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_912_diff__eq__diff__eq,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D ) )
=> ( ( A = B )
= ( C2 = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_913_diff__mono,axiom,
! [A: real,B: real,D: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D @ C2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_914_diff__mono,axiom,
! [A: int,B: int,D: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_915_diff__left__mono,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C2 @ A ) @ ( minus_minus_real @ C2 @ B ) ) ) ).
% diff_left_mono
thf(fact_916_diff__left__mono,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).
% diff_left_mono
thf(fact_917_diff__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ C2 ) ) ) ).
% diff_right_mono
thf(fact_918_diff__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).
% diff_right_mono
thf(fact_919_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C2 @ D ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C2 @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_920_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C2 @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_921_diff__right__commute,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_922_diff__right__commute,axiom,
! [A: real,C2: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C2 ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_923_diff__right__commute,axiom,
! [A: int,C2: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_924_one__reorient,axiom,
! [X2: real] :
( ( one_one_real = X2 )
= ( X2 = one_one_real ) ) ).
% one_reorient
thf(fact_925_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_926_one__reorient,axiom,
! [X2: extended_ereal] :
( ( one_on4623092294121504201_ereal = X2 )
= ( X2 = one_on4623092294121504201_ereal ) ) ).
% one_reorient
thf(fact_927_one__reorient,axiom,
! [X2: int] :
( ( one_one_int = X2 )
= ( X2 = one_one_int ) ) ).
% one_reorient
thf(fact_928_ereal__minus__mono,axiom,
! [A3: extended_ereal,B3: extended_ereal,D3: extended_ereal,C4: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A3 @ B3 )
=> ( ( ord_le1083603963089353582_ereal @ D3 @ C4 )
=> ( ord_le1083603963089353582_ereal @ ( minus_2816186181549245109_ereal @ A3 @ C4 ) @ ( minus_2816186181549245109_ereal @ B3 @ D3 ) ) ) ) ).
% ereal_minus_mono
thf(fact_929_ereal__complete__Inf,axiom,
! [S2: set_Extended_ereal] :
? [X3: extended_ereal] :
( ! [Xa2: extended_ereal] :
( ( member2350847679896131959_ereal @ Xa2 @ S2 )
=> ( ord_le1083603963089353582_ereal @ X3 @ Xa2 ) )
& ! [Z3: extended_ereal] :
( ! [Xa3: extended_ereal] :
( ( member2350847679896131959_ereal @ Xa3 @ S2 )
=> ( ord_le1083603963089353582_ereal @ Z3 @ Xa3 ) )
=> ( ord_le1083603963089353582_ereal @ Z3 @ X3 ) ) ) ).
% ereal_complete_Inf
thf(fact_930_ereal__complete__Sup,axiom,
! [S2: set_Extended_ereal] :
? [X3: extended_ereal] :
( ! [Xa2: extended_ereal] :
( ( member2350847679896131959_ereal @ Xa2 @ S2 )
=> ( ord_le1083603963089353582_ereal @ Xa2 @ X3 ) )
& ! [Z3: extended_ereal] :
( ! [Xa3: extended_ereal] :
( ( member2350847679896131959_ereal @ Xa3 @ S2 )
=> ( ord_le1083603963089353582_ereal @ Xa3 @ Z3 ) )
=> ( ord_le1083603963089353582_ereal @ X3 @ Z3 ) ) ) ).
% ereal_complete_Sup
thf(fact_931_diff__diff__commute__ereal,axiom,
! [X2: extended_ereal,Y: extended_ereal,Z2: extended_ereal] :
( ( minus_2816186181549245109_ereal @ ( minus_2816186181549245109_ereal @ X2 @ Y ) @ Z2 )
= ( minus_2816186181549245109_ereal @ ( minus_2816186181549245109_ereal @ X2 @ Z2 ) @ Y ) ) ).
% diff_diff_commute_ereal
thf(fact_932_ereal__diff__le__mono__left,axiom,
! [X2: extended_ereal,Z2: extended_ereal,Y: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X2 @ Z2 )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ Y )
=> ( ord_le1083603963089353582_ereal @ ( minus_2816186181549245109_ereal @ X2 @ Y ) @ Z2 ) ) ) ).
% ereal_diff_le_mono_left
thf(fact_933_ereal__diff__positive,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( minus_2816186181549245109_ereal @ B @ A ) ) ) ).
% ereal_diff_positive
thf(fact_934_ereal__diff__le__self,axiom,
! [Y: extended_ereal,X2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ Y )
=> ( ord_le1083603963089353582_ereal @ ( minus_2816186181549245109_ereal @ X2 @ Y ) @ X2 ) ) ).
% ereal_diff_le_self
thf(fact_935_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A2: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_936_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A2: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_937_verit__comp__simplify1_I2_J,axiom,
! [A: extended_ereal] : ( ord_le1083603963089353582_ereal @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_938_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_939_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_940_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_941_diff__left__imp__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ A @ C2 ) )
=> ( B = C2 ) ) ).
% diff_left_imp_eq
thf(fact_942_diff__left__imp__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ A @ C2 ) )
=> ( B = C2 ) ) ).
% diff_left_imp_eq
thf(fact_943_verit__la__disequality,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( A = B )
| ~ ( ord_le1083603963089353582_ereal @ A @ B )
| ~ ( ord_le1083603963089353582_ereal @ B @ A ) ) ).
% verit_la_disequality
thf(fact_944_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_945_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_946_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_947_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_948_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_949_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_950_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_951_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_952_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_953_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_954_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_955_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_956_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_957_add__le__add__imp__diff__le,axiom,
! [I: real,K: real,N: real,J: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
=> ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
=> ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_958_add__le__add__imp__diff__le,axiom,
! [I: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_959_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_960_add__le__imp__le__diff,axiom,
! [I: real,K: real,N: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
=> ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_961_add__le__imp__le__diff,axiom,
! [I: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_962_ereal__diff__add__assoc2,axiom,
! [X2: extended_ereal,Y: extended_ereal,Z2: extended_ereal] :
( ( minus_2816186181549245109_ereal @ ( plus_p7876563987511257093_ereal @ X2 @ Y ) @ Z2 )
= ( plus_p7876563987511257093_ereal @ ( minus_2816186181549245109_ereal @ X2 @ Z2 ) @ Y ) ) ).
% ereal_diff_add_assoc2
thf(fact_963_diff__add__eq__ereal,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ ( minus_2816186181549245109_ereal @ A @ B ) @ C2 )
= ( minus_2816186181549245109_ereal @ ( plus_p7876563987511257093_ereal @ A @ C2 ) @ B ) ) ).
% diff_add_eq_ereal
thf(fact_964_add__diff__eq__ereal,axiom,
! [X2: extended_ereal,Y: extended_ereal,Z2: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ X2 @ ( minus_2816186181549245109_ereal @ Y @ Z2 ) )
= ( minus_2816186181549245109_ereal @ ( plus_p7876563987511257093_ereal @ X2 @ Y ) @ Z2 ) ) ).
% add_diff_eq_ereal
thf(fact_965_ereal__uminus__le__reorder,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ A ) @ B )
= ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ B ) @ A ) ) ).
% ereal_uminus_le_reorder
thf(fact_966_ereal__le__real,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ! [Z: real] :
( ( ord_le1083603963089353582_ereal @ X2 @ ( extended_ereal2 @ Z ) )
=> ( ord_le1083603963089353582_ereal @ Y @ ( extended_ereal2 @ Z ) ) )
=> ( ord_le1083603963089353582_ereal @ Y @ X2 ) ) ).
% ereal_le_real
thf(fact_967_neq__PInf__trans,axiom,
! [Y: extended_ereal,X2: extended_ereal] :
( ( Y != extend1530274965995635425_ereal )
=> ( ( ord_le1083603963089353582_ereal @ X2 @ Y )
=> ( X2 != extend1530274965995635425_ereal ) ) ) ).
% neq_PInf_trans
thf(fact_968_ereal__infty__less__eq2_I1_J,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( A = extend1530274965995635425_ereal )
=> ( B = extend1530274965995635425_ereal ) ) ) ).
% ereal_infty_less_eq2(1)
thf(fact_969_ereal__less__eq_I1_J,axiom,
! [X2: extended_ereal] : ( ord_le1083603963089353582_ereal @ X2 @ extend1530274965995635425_ereal ) ).
% ereal_less_eq(1)
thf(fact_970_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_971_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_972_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_973_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_974_zero__neq__one,axiom,
zero_z2744965634713055877_ereal != one_on4623092294121504201_ereal ).
% zero_neq_one
thf(fact_975_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_976_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_977_le__imp__neg__le,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% le_imp_neg_le
thf(fact_978_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_979_minus__le__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_980_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_981_le__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% le_minus_iff
thf(fact_982_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_983_add__le__imp__le__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_984_add__le__imp__le__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_985_add__le__imp__le__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_986_add__le__imp__le__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_987_add__le__imp__le__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_988_add__le__imp__le__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_989_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
? [C: nat] :
( B2
= ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% le_iff_add
thf(fact_990_add__right__mono,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ A @ C2 ) @ ( plus_p7876563987511257093_ereal @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_991_add__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_992_add__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_993_add__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_994_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_995_add__left__mono,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ C2 @ A ) @ ( plus_p7876563987511257093_ereal @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_996_add__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_997_add__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_998_add__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_999_add__mono,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal,D: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ord_le1083603963089353582_ereal @ C2 @ D )
=> ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ A @ C2 ) @ ( plus_p7876563987511257093_ereal @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1000_add__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1001_add__mono,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1002_add__mono,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1003_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: extended_ereal,J: extended_ereal,K: extended_ereal,L: extended_ereal] :
( ( ( ord_le1083603963089353582_ereal @ I @ J )
& ( ord_le1083603963089353582_ereal @ K @ L ) )
=> ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ I @ K ) @ ( plus_p7876563987511257093_ereal @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1004_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1005_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1006_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1007_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: extended_ereal,J: extended_ereal,K: extended_ereal,L: extended_ereal] :
( ( ( I = J )
& ( ord_le1083603963089353582_ereal @ K @ L ) )
=> ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ I @ K ) @ ( plus_p7876563987511257093_ereal @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1008_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1009_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1010_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1011_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: extended_ereal,J: extended_ereal,K: extended_ereal,L: extended_ereal] :
( ( ( ord_le1083603963089353582_ereal @ I @ J )
& ( K = L ) )
=> ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ I @ K ) @ ( plus_p7876563987511257093_ereal @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1012_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1013_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1014_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1015_verit__comp__simplify1_I3_J,axiom,
! [B5: extended_ereal,A5: extended_ereal] :
( ( ~ ( ord_le1083603963089353582_ereal @ B5 @ A5 ) )
= ( ord_le1188267648640031866_ereal @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1016_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
= ( ord_less_nat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1017_verit__comp__simplify1_I3_J,axiom,
! [B5: real,A5: real] :
( ( ~ ( ord_less_eq_real @ B5 @ A5 ) )
= ( ord_less_real @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1018_verit__comp__simplify1_I3_J,axiom,
! [B5: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B5 @ A5 ) )
= ( ord_less_int @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1019_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_1020_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1021_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_1022_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_1023_minus__diff__commute,axiom,
! [B: real,A: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
= ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_1024_minus__diff__commute,axiom,
! [B: int,A: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
= ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_1025_diff__diff__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_1026_diff__diff__eq,axiom,
! [A: real,B: real,C2: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C2 )
= ( minus_minus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_1027_diff__diff__eq,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_1028_add__implies__diff,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1029_add__implies__diff,axiom,
! [C2: real,B: real,A: real] :
( ( ( plus_plus_real @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_real @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1030_add__implies__diff,axiom,
! [C2: int,B: int,A: int] :
( ( ( plus_plus_int @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1031_diff__add__eq__diff__diff__swap,axiom,
! [A: real,B: real,C2: real] :
( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C2 ) )
= ( minus_minus_real @ ( minus_minus_real @ A @ C2 ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1032_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1033_diff__add__eq,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C2 )
= ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ).
% diff_add_eq
thf(fact_1034_diff__add__eq,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).
% diff_add_eq
thf(fact_1035_diff__diff__eq2,axiom,
! [A: real,B: real,C2: real] :
( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C2 ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ).
% diff_diff_eq2
thf(fact_1036_diff__diff__eq2,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).
% diff_diff_eq2
thf(fact_1037_add__diff__eq,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C2 ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C2 ) ) ).
% add_diff_eq
thf(fact_1038_add__diff__eq,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% add_diff_eq
thf(fact_1039_eq__diff__eq,axiom,
! [A: real,C2: real,B: real] :
( ( A
= ( minus_minus_real @ C2 @ B ) )
= ( ( plus_plus_real @ A @ B )
= C2 ) ) ).
% eq_diff_eq
thf(fact_1040_eq__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( A
= ( minus_minus_int @ C2 @ B ) )
= ( ( plus_plus_int @ A @ B )
= C2 ) ) ).
% eq_diff_eq
thf(fact_1041_diff__eq__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ( minus_minus_real @ A @ B )
= C2 )
= ( A
= ( plus_plus_real @ C2 @ B ) ) ) ).
% diff_eq_eq
thf(fact_1042_diff__eq__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ( minus_minus_int @ A @ B )
= C2 )
= ( A
= ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_eq_eq
thf(fact_1043_group__cancel_Osub1,axiom,
! [A3: real,K: real,A: real,B: real] :
( ( A3
= ( plus_plus_real @ K @ A ) )
=> ( ( minus_minus_real @ A3 @ B )
= ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_1044_group__cancel_Osub1,axiom,
! [A3: int,K: int,A: int,B: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A3 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_1045_add__diff__add,axiom,
! [A: real,C2: real,B: real,D: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D ) )
= ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C2 @ D ) ) ) ).
% add_diff_add
thf(fact_1046_add__diff__add,axiom,
! [A: int,C2: int,B: int,D: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) )
= ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C2 @ D ) ) ) ).
% add_diff_add
thf(fact_1047_diff__strict__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_1048_diff__strict__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_1049_diff__strict__left__mono,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_1050_ereal__diff__nonpos,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ( A = extend1530274965995635425_ereal )
=> ( B != extend1530274965995635425_ereal ) )
=> ( ( ( A
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( B
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) )
=> ( ord_le1083603963089353582_ereal @ ( minus_2816186181549245109_ereal @ A @ B ) @ zero_z2744965634713055877_ereal ) ) ) ) ).
% ereal_diff_nonpos
thf(fact_1051_ereal__mono__minus__cancel,axiom,
! [C2: extended_ereal,A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( minus_2816186181549245109_ereal @ C2 @ A ) @ ( minus_2816186181549245109_ereal @ C2 @ B ) )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ C2 )
=> ( ( ord_le1188267648640031866_ereal @ C2 @ extend1530274965995635425_ereal )
=> ( ord_le1083603963089353582_ereal @ B @ A ) ) ) ) ).
% ereal_mono_minus_cancel
thf(fact_1052_ereal__minus__le,axiom,
! [Y: extended_ereal,X2: extended_ereal,Z2: extended_ereal] :
( ( ( abs_ab7465543570706387889_ereal @ Y )
!= extend1530274965995635425_ereal )
=> ( ( ord_le1083603963089353582_ereal @ ( minus_2816186181549245109_ereal @ X2 @ Y ) @ Z2 )
= ( ord_le1083603963089353582_ereal @ X2 @ ( plus_p7876563987511257093_ereal @ Z2 @ Y ) ) ) ) ).
% ereal_minus_le
thf(fact_1053_ereal__le__minus,axiom,
! [Y: extended_ereal,X2: extended_ereal,Z2: extended_ereal] :
( ( ( abs_ab7465543570706387889_ereal @ Y )
!= extend1530274965995635425_ereal )
=> ( ( ord_le1083603963089353582_ereal @ X2 @ ( minus_2816186181549245109_ereal @ Z2 @ Y ) )
= ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ X2 @ Y ) @ Z2 ) ) ) ).
% ereal_le_minus
thf(fact_1054_ereal__minus__le__iff,axiom,
! [X2: extended_ereal,Y: extended_ereal,Z2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( minus_2816186181549245109_ereal @ X2 @ Y ) @ Z2 )
= ( ( ( Y
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Z2 = extend1530274965995635425_ereal ) )
& ( ( Y = extend1530274965995635425_ereal )
=> ( ( X2 = extend1530274965995635425_ereal )
=> ( Z2 = extend1530274965995635425_ereal ) ) )
& ( ( ( abs_ab7465543570706387889_ereal @ Y )
!= extend1530274965995635425_ereal )
=> ( ord_le1083603963089353582_ereal @ X2 @ ( plus_p7876563987511257093_ereal @ Z2 @ Y ) ) ) ) ) ).
% ereal_minus_le_iff
thf(fact_1055_ereal__le__minus__iff,axiom,
! [X2: extended_ereal,Z2: extended_ereal,Y: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X2 @ ( minus_2816186181549245109_ereal @ Z2 @ Y ) )
= ( ( ( Y = extend1530274965995635425_ereal )
=> ( ( Z2 != extend1530274965995635425_ereal )
=> ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) )
& ( ( ( abs_ab7465543570706387889_ereal @ Y )
!= extend1530274965995635425_ereal )
=> ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ X2 @ Y ) @ Z2 ) ) ) ) ).
% ereal_le_minus_iff
thf(fact_1056_ereal__minus__diff__eq,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( ( X2 = extend1530274965995635425_ereal )
=> ( Y != extend1530274965995635425_ereal ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Y
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) )
=> ( ( uminus27091377158695749_ereal @ ( minus_2816186181549245109_ereal @ X2 @ Y ) )
= ( minus_2816186181549245109_ereal @ Y @ X2 ) ) ) ) ).
% ereal_minus_diff_eq
thf(fact_1057_ereal__minus__eq__minus__iff,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( ( minus_2816186181549245109_ereal @ A @ B )
= ( minus_2816186181549245109_ereal @ A @ C2 ) )
= ( ( B = C2 )
| ( A = extend1530274965995635425_ereal )
| ( ( A
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
& ( B
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
& ( C2
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ).
% ereal_minus_eq_minus_iff
thf(fact_1058_ereal__minus__eq__PInfty__iff,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( ( minus_2816186181549245109_ereal @ X2 @ Y )
= extend1530274965995635425_ereal )
= ( ( Y
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
| ( X2 = extend1530274965995635425_ereal ) ) ) ).
% ereal_minus_eq_PInfty_iff
thf(fact_1059_abs__ereal_Osimps_I1_J,axiom,
! [R: real] :
( ( abs_ab7465543570706387889_ereal @ ( extended_ereal2 @ R ) )
= ( extended_ereal2 @ ( abs_abs_real @ R ) ) ) ).
% abs_ereal.simps(1)
thf(fact_1060_ereal__add__uminus__conv__diff,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ ( uminus27091377158695749_ereal @ X2 ) @ Y )
= ( minus_2816186181549245109_ereal @ Y @ X2 ) ) ).
% ereal_add_uminus_conv_diff
thf(fact_1061_minus__ereal__def,axiom,
( minus_2816186181549245109_ereal
= ( ^ [X: extended_ereal,Y9: extended_ereal] : ( plus_p7876563987511257093_ereal @ X @ ( uminus27091377158695749_ereal @ Y9 ) ) ) ) ).
% minus_ereal_def
thf(fact_1062_ereal__top,axiom,
! [X2: extended_ereal] :
( ! [B6: real] : ( ord_le1083603963089353582_ereal @ ( extended_ereal2 @ B6 ) @ X2 )
=> ( X2 = extend1530274965995635425_ereal ) ) ).
% ereal_top
thf(fact_1063_ereal__less__eq_I2_J,axiom,
! [X2: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ X2 ) ).
% ereal_less_eq(2)
thf(fact_1064_ereal__infty__less__eq2_I2_J,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( B
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( A
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ).
% ereal_infty_less_eq2(2)
thf(fact_1065_ereal__diff__gr0,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ ( minus_2816186181549245109_ereal @ B @ A ) ) ) ).
% ereal_diff_gr0
thf(fact_1066_ereal__abs__leI,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X2 @ Y )
=> ( ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ X2 ) @ Y )
=> ( ord_le1083603963089353582_ereal @ ( abs_ab7465543570706387889_ereal @ X2 ) @ Y ) ) ) ).
% ereal_abs_leI
thf(fact_1067_ereal__mult__right__mono,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ C2 )
=> ( ord_le1083603963089353582_ereal @ ( times_7703590493115627913_ereal @ A @ C2 ) @ ( times_7703590493115627913_ereal @ B @ C2 ) ) ) ) ).
% ereal_mult_right_mono
thf(fact_1068_ereal__mult__left__mono,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ C2 )
=> ( ord_le1083603963089353582_ereal @ ( times_7703590493115627913_ereal @ C2 @ A ) @ ( times_7703590493115627913_ereal @ C2 @ B ) ) ) ) ).
% ereal_mult_left_mono
thf(fact_1069_ereal__zero__le__0__iff,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( times_7703590493115627913_ereal @ A @ B ) )
= ( ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A )
& ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ B ) )
| ( ( ord_le1083603963089353582_ereal @ A @ zero_z2744965634713055877_ereal )
& ( ord_le1083603963089353582_ereal @ B @ zero_z2744965634713055877_ereal ) ) ) ) ).
% ereal_zero_le_0_iff
thf(fact_1070_ereal__mult__le__0__iff,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( times_7703590493115627913_ereal @ A @ B ) @ zero_z2744965634713055877_ereal )
= ( ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A )
& ( ord_le1083603963089353582_ereal @ B @ zero_z2744965634713055877_ereal ) )
| ( ( ord_le1083603963089353582_ereal @ A @ zero_z2744965634713055877_ereal )
& ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ B ) ) ) ) ).
% ereal_mult_le_0_iff
thf(fact_1071_ereal__mult__mono_H,axiom,
! [A: extended_ereal,C2: extended_ereal,B: extended_ereal,D: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ C2 )
=> ( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ord_le1083603963089353582_ereal @ C2 @ D )
=> ( ord_le1083603963089353582_ereal @ ( times_7703590493115627913_ereal @ A @ C2 ) @ ( times_7703590493115627913_ereal @ B @ D ) ) ) ) ) ) ).
% ereal_mult_mono'
thf(fact_1072_ereal__mult__mono,axiom,
! [B: extended_ereal,C2: extended_ereal,A: extended_ereal,D: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ B )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ C2 )
=> ( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ord_le1083603963089353582_ereal @ C2 @ D )
=> ( ord_le1083603963089353582_ereal @ ( times_7703590493115627913_ereal @ A @ C2 ) @ ( times_7703590493115627913_ereal @ B @ D ) ) ) ) ) ) ).
% ereal_mult_mono
thf(fact_1073_ereal__0__le__mult,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ B )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( times_7703590493115627913_ereal @ A @ B ) ) ) ) ).
% ereal_0_le_mult
thf(fact_1074_ereal__le__add__self,axiom,
! [Y: extended_ereal,X2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ Y )
=> ( ord_le1083603963089353582_ereal @ X2 @ ( plus_p7876563987511257093_ereal @ X2 @ Y ) ) ) ).
% ereal_le_add_self
thf(fact_1075_ereal__le__add__mono1,axiom,
! [X2: extended_ereal,Y: extended_ereal,Z2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X2 @ Y )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ Z2 )
=> ( ord_le1083603963089353582_ereal @ X2 @ ( plus_p7876563987511257093_ereal @ Y @ Z2 ) ) ) ) ).
% ereal_le_add_mono1
thf(fact_1076_ereal__le__add__mono2,axiom,
! [X2: extended_ereal,Z2: extended_ereal,Y: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X2 @ Z2 )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ Y )
=> ( ord_le1083603963089353582_ereal @ X2 @ ( plus_p7876563987511257093_ereal @ Y @ Z2 ) ) ) ) ).
% ereal_le_add_mono2
thf(fact_1077_ereal__le__add__self2,axiom,
! [Y: extended_ereal,X2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ Y )
=> ( ord_le1083603963089353582_ereal @ X2 @ ( plus_p7876563987511257093_ereal @ Y @ X2 ) ) ) ).
% ereal_le_add_self2
thf(fact_1078_ereal__add__nonneg__eq__0__iff,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ B )
=> ( ( ( plus_p7876563987511257093_ereal @ A @ B )
= zero_z2744965634713055877_ereal )
= ( ( A = zero_z2744965634713055877_ereal )
& ( B = zero_z2744965634713055877_ereal ) ) ) ) ) ).
% ereal_add_nonneg_eq_0_iff
thf(fact_1079_abs__ereal__pos,axiom,
! [X2: extended_ereal] : ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( abs_ab7465543570706387889_ereal @ X2 ) ) ).
% abs_ereal_pos
thf(fact_1080_ereal__le__distrib,axiom,
! [C2: extended_ereal,A: extended_ereal,B: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( times_7703590493115627913_ereal @ C2 @ ( plus_p7876563987511257093_ereal @ A @ B ) ) @ ( plus_p7876563987511257093_ereal @ ( times_7703590493115627913_ereal @ C2 @ A ) @ ( times_7703590493115627913_ereal @ C2 @ B ) ) ) ).
% ereal_le_distrib
thf(fact_1081_ereal__abs__add,axiom,
! [A: extended_ereal,B: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( abs_ab7465543570706387889_ereal @ ( plus_p7876563987511257093_ereal @ A @ B ) ) @ ( plus_p7876563987511257093_ereal @ ( abs_ab7465543570706387889_ereal @ A ) @ ( abs_ab7465543570706387889_ereal @ B ) ) ) ).
% ereal_abs_add
thf(fact_1082_abs__real__def,axiom,
( abs_abs_real
= ( ^ [A2: real] : ( if_real @ ( ord_less_real @ A2 @ zero_zero_real ) @ ( uminus_uminus_real @ A2 ) @ A2 ) ) ) ).
% abs_real_def
thf(fact_1083_ereal__diff__eq__MInfty__iff,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( ( minus_2816186181549245109_ereal @ X2 @ Y )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
= ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
& ( Y
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) )
| ( ( Y = extend1530274965995635425_ereal )
& ( ( abs_ab7465543570706387889_ereal @ X2 )
!= extend1530274965995635425_ereal ) ) ) ) ).
% ereal_diff_eq_MInfty_iff
thf(fact_1084_ereal__diff__eq__0__iff,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( ( abs_ab7465543570706387889_ereal @ A )
= extend1530274965995635425_ereal )
=> ( ( abs_ab7465543570706387889_ereal @ B )
!= extend1530274965995635425_ereal ) )
=> ( ( ( minus_2816186181549245109_ereal @ A @ B )
= zero_z2744965634713055877_ereal )
= ( A = B ) ) ) ).
% ereal_diff_eq_0_iff
thf(fact_1085_ereal__bot,axiom,
! [X2: extended_ereal] :
( ! [B6: real] : ( ord_le1083603963089353582_ereal @ X2 @ ( extended_ereal2 @ B6 ) )
=> ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ).
% ereal_bot
thf(fact_1086_ereal__diff__add__eq__diff__diff__swap,axiom,
! [Y: extended_ereal,X2: extended_ereal,Z2: extended_ereal] :
( ( ( abs_ab7465543570706387889_ereal @ Y )
!= extend1530274965995635425_ereal )
=> ( ( minus_2816186181549245109_ereal @ X2 @ ( plus_p7876563987511257093_ereal @ Y @ Z2 ) )
= ( minus_2816186181549245109_ereal @ ( minus_2816186181549245109_ereal @ X2 @ Y ) @ Z2 ) ) ) ).
% ereal_diff_add_eq_diff_diff_swap
thf(fact_1087_ereal__diff__add__inverse,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( ( abs_ab7465543570706387889_ereal @ X2 )
!= extend1530274965995635425_ereal )
=> ( ( minus_2816186181549245109_ereal @ ( plus_p7876563987511257093_ereal @ X2 @ Y ) @ X2 )
= Y ) ) ).
% ereal_diff_add_inverse
thf(fact_1088_ereal__minus__minus,axiom,
! [Y: extended_ereal,Z2: extended_ereal,X2: extended_ereal] :
( ( ( ( abs_ab7465543570706387889_ereal @ Y )
= extend1530274965995635425_ereal )
=> ( ( abs_ab7465543570706387889_ereal @ Z2 )
!= extend1530274965995635425_ereal ) )
=> ( ( minus_2816186181549245109_ereal @ X2 @ ( minus_2816186181549245109_ereal @ Y @ Z2 ) )
= ( minus_2816186181549245109_ereal @ ( plus_p7876563987511257093_ereal @ X2 @ Z2 ) @ Y ) ) ) ).
% ereal_minus_minus
thf(fact_1089_ereal__eq__minus,axiom,
! [Y: extended_ereal,X2: extended_ereal,Z2: extended_ereal] :
( ( ( abs_ab7465543570706387889_ereal @ Y )
!= extend1530274965995635425_ereal )
=> ( ( X2
= ( minus_2816186181549245109_ereal @ Z2 @ Y ) )
= ( ( plus_p7876563987511257093_ereal @ X2 @ Y )
= Z2 ) ) ) ).
% ereal_eq_minus
thf(fact_1090_not__MInfty__nonneg,axiom,
! [X2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ X2 )
=> ( X2
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ).
% not_MInfty_nonneg
thf(fact_1091_ereal__le__less,axiom,
! [Y: real,A: extended_ereal,X2: real] :
( ( ord_le1083603963089353582_ereal @ ( extended_ereal2 @ Y ) @ A )
=> ( ( ord_less_real @ X2 @ Y )
=> ( ord_le1188267648640031866_ereal @ ( extended_ereal2 @ X2 ) @ A ) ) ) ).
% ereal_le_less
thf(fact_1092_le__ereal__less,axiom,
! [A: extended_ereal,X2: real,Y: real] :
( ( ord_le1083603963089353582_ereal @ A @ ( extended_ereal2 @ X2 ) )
=> ( ( ord_less_real @ X2 @ Y )
=> ( ord_le1188267648640031866_ereal @ A @ ( extended_ereal2 @ Y ) ) ) ) ).
% le_ereal_less
thf(fact_1093_ereal__add__le__add__iff,axiom,
! [C2: extended_ereal,A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ C2 @ A ) @ ( plus_p7876563987511257093_ereal @ C2 @ B ) )
= ( ( ord_le1083603963089353582_ereal @ A @ B )
| ( C2 = extend1530274965995635425_ereal )
| ( ( C2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
& ( A != extend1530274965995635425_ereal )
& ( B != extend1530274965995635425_ereal ) ) ) ) ).
% ereal_add_le_add_iff
thf(fact_1094_ereal__add__le__add__iff2,axiom,
! [A: extended_ereal,C2: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( plus_p7876563987511257093_ereal @ A @ C2 ) @ ( plus_p7876563987511257093_ereal @ B @ C2 ) )
= ( ( ord_le1083603963089353582_ereal @ A @ B )
| ( C2 = extend1530274965995635425_ereal )
| ( ( C2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
& ( A != extend1530274965995635425_ereal )
& ( B != extend1530274965995635425_ereal ) ) ) ) ).
% ereal_add_le_add_iff2
thf(fact_1095_ereal__le__epsilon,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ! [E: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ E )
=> ( ord_le1083603963089353582_ereal @ X2 @ ( plus_p7876563987511257093_ereal @ Y @ E ) ) )
=> ( ord_le1083603963089353582_ereal @ X2 @ Y ) ) ).
% ereal_le_epsilon
thf(fact_1096_ereal__right__distrib,axiom,
! [A: extended_ereal,B: extended_ereal,R: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ B )
=> ( ( times_7703590493115627913_ereal @ R @ ( plus_p7876563987511257093_ereal @ A @ B ) )
= ( plus_p7876563987511257093_ereal @ ( times_7703590493115627913_ereal @ R @ A ) @ ( times_7703590493115627913_ereal @ R @ B ) ) ) ) ) ).
% ereal_right_distrib
thf(fact_1097_ereal__left__distrib,axiom,
! [A: extended_ereal,B: extended_ereal,R: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ B )
=> ( ( times_7703590493115627913_ereal @ ( plus_p7876563987511257093_ereal @ A @ B ) @ R )
= ( plus_p7876563987511257093_ereal @ ( times_7703590493115627913_ereal @ A @ R ) @ ( times_7703590493115627913_ereal @ B @ R ) ) ) ) ) ).
% ereal_left_distrib
thf(fact_1098_abs__ereal_Oelims,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( ( abs_ab7465543570706387889_ereal @ X2 )
= Y )
=> ( ! [R2: real] :
( ( X2
= ( extended_ereal2 @ R2 ) )
=> ( Y
!= ( extended_ereal2 @ ( abs_abs_real @ R2 ) ) ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Y != extend1530274965995635425_ereal ) )
=> ~ ( ( X2 = extend1530274965995635425_ereal )
=> ( Y != extend1530274965995635425_ereal ) ) ) ) ) ).
% abs_ereal.elims
thf(fact_1099_ereal__minus__less__iff,axiom,
! [X2: extended_ereal,Y: extended_ereal,Z2: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ ( minus_2816186181549245109_ereal @ X2 @ Y ) @ Z2 )
= ( ( Y
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
& ( ( Y = extend1530274965995635425_ereal )
=> ( ( X2 != extend1530274965995635425_ereal )
& ( Z2
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) )
& ( ( Y != extend1530274965995635425_ereal )
=> ( ord_le1188267648640031866_ereal @ X2 @ ( plus_p7876563987511257093_ereal @ Z2 @ Y ) ) ) ) ) ).
% ereal_minus_less_iff
thf(fact_1100_ereal__distrib__minus__right,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( ( A != extend1530274965995635425_ereal )
| ( B != extend1530274965995635425_ereal ) )
=> ( ( ( A
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
| ( B
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) )
=> ( ( ( abs_ab7465543570706387889_ereal @ C2 )
!= extend1530274965995635425_ereal )
=> ( ( times_7703590493115627913_ereal @ ( minus_2816186181549245109_ereal @ A @ B ) @ C2 )
= ( minus_2816186181549245109_ereal @ ( times_7703590493115627913_ereal @ A @ C2 ) @ ( times_7703590493115627913_ereal @ B @ C2 ) ) ) ) ) ) ).
% ereal_distrib_minus_right
thf(fact_1101_ereal__distrib__minus__left,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( ( A != extend1530274965995635425_ereal )
| ( B != extend1530274965995635425_ereal ) )
=> ( ( ( A
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
| ( B
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) )
=> ( ( ( abs_ab7465543570706387889_ereal @ C2 )
!= extend1530274965995635425_ereal )
=> ( ( times_7703590493115627913_ereal @ C2 @ ( minus_2816186181549245109_ereal @ A @ B ) )
= ( minus_2816186181549245109_ereal @ ( times_7703590493115627913_ereal @ C2 @ A ) @ ( times_7703590493115627913_ereal @ C2 @ B ) ) ) ) ) ) ).
% ereal_distrib_minus_left
thf(fact_1102_ereal__eq__minus__iff,axiom,
! [X2: extended_ereal,Z2: extended_ereal,Y: extended_ereal] :
( ( X2
= ( minus_2816186181549245109_ereal @ Z2 @ Y ) )
= ( ( ( ( abs_ab7465543570706387889_ereal @ Y )
!= extend1530274965995635425_ereal )
=> ( ( plus_p7876563987511257093_ereal @ X2 @ Y )
= Z2 ) )
& ( ( Y
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( X2 = extend1530274965995635425_ereal ) )
& ( ( Y = extend1530274965995635425_ereal )
=> ( ( Z2 = extend1530274965995635425_ereal )
=> ( X2 = extend1530274965995635425_ereal ) ) )
& ( ( Y = extend1530274965995635425_ereal )
=> ( ( Z2 != extend1530274965995635425_ereal )
=> ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ) ).
% ereal_eq_minus_iff
thf(fact_1103_ereal__between_I1_J,axiom,
! [X2: extended_ereal,E2: extended_ereal] :
( ( ( abs_ab7465543570706387889_ereal @ X2 )
!= extend1530274965995635425_ereal )
=> ( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ E2 )
=> ( ord_le1188267648640031866_ereal @ ( minus_2816186181549245109_ereal @ X2 @ E2 ) @ X2 ) ) ) ).
% ereal_between(1)
thf(fact_1104_ereal__less__minus,axiom,
! [Y: extended_ereal,X2: extended_ereal,Z2: extended_ereal] :
( ( ( abs_ab7465543570706387889_ereal @ Y )
!= extend1530274965995635425_ereal )
=> ( ( ord_le1188267648640031866_ereal @ X2 @ ( minus_2816186181549245109_ereal @ Z2 @ Y ) )
= ( ord_le1188267648640031866_ereal @ ( plus_p7876563987511257093_ereal @ X2 @ Y ) @ Z2 ) ) ) ).
% ereal_less_minus
thf(fact_1105_ereal__minus__less,axiom,
! [Y: extended_ereal,X2: extended_ereal,Z2: extended_ereal] :
( ( ( abs_ab7465543570706387889_ereal @ Y )
!= extend1530274965995635425_ereal )
=> ( ( ord_le1188267648640031866_ereal @ ( minus_2816186181549245109_ereal @ X2 @ Y ) @ Z2 )
= ( ord_le1188267648640031866_ereal @ X2 @ ( plus_p7876563987511257093_ereal @ Z2 @ Y ) ) ) ) ).
% ereal_minus_less
thf(fact_1106_ereal__le__epsilon2,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_le1083603963089353582_ereal @ X2 @ ( plus_p7876563987511257093_ereal @ Y @ ( extended_ereal2 @ E ) ) ) )
=> ( ord_le1083603963089353582_ereal @ X2 @ Y ) ) ).
% ereal_le_epsilon2
thf(fact_1107_ereal__add__strict__mono,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal,D: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( A != extend1530274965995635425_ereal )
=> ( ( ord_le1188267648640031866_ereal @ C2 @ D )
=> ( ord_le1188267648640031866_ereal @ ( plus_p7876563987511257093_ereal @ A @ C2 ) @ ( plus_p7876563987511257093_ereal @ B @ D ) ) ) ) ) ) ).
% ereal_add_strict_mono
thf(fact_1108_ereal__pos__distrib,axiom,
! [C2: extended_ereal,A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ C2 )
=> ( ( C2 != extend1530274965995635425_ereal )
=> ( ( times_7703590493115627913_ereal @ C2 @ ( plus_p7876563987511257093_ereal @ A @ B ) )
= ( plus_p7876563987511257093_ereal @ ( times_7703590493115627913_ereal @ C2 @ A ) @ ( times_7703590493115627913_ereal @ C2 @ B ) ) ) ) ) ).
% ereal_pos_distrib
thf(fact_1109_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1110_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1111_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1112_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1113_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1114_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1115_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1116_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1117_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1118_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1119_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1120_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1121_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1122_abs__ereal__one,axiom,
( ( abs_ab7465543570706387889_ereal @ one_on4623092294121504201_ereal )
= one_on4623092294121504201_ereal ) ).
% abs_ereal_one
thf(fact_1123_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1124_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1125_ereal__eq__1_I1_J,axiom,
! [R: real] :
( ( ( extended_ereal2 @ R )
= one_on4623092294121504201_ereal )
= ( R = one_one_real ) ) ).
% ereal_eq_1(1)
thf(fact_1126_ereal__eq__1_I2_J,axiom,
! [R: real] :
( ( one_on4623092294121504201_ereal
= ( extended_ereal2 @ R ) )
= ( R = one_one_real ) ) ).
% ereal_eq_1(2)
thf(fact_1127_ereal__less__eq_I3_J,axiom,
! [R: real,P2: real] :
( ( ord_le1083603963089353582_ereal @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P2 ) )
= ( ord_less_eq_real @ R @ P2 ) ) ).
% ereal_less_eq(3)
thf(fact_1128_real__ereal__1,axiom,
( ( extend2982805604970551563_ereal @ one_on4623092294121504201_ereal )
= one_one_real ) ).
% real_ereal_1
thf(fact_1129_ereal__minus_I1_J,axiom,
! [R: real,P2: real] :
( ( minus_2816186181549245109_ereal @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P2 ) )
= ( extended_ereal2 @ ( minus_minus_real @ R @ P2 ) ) ) ).
% ereal_minus(1)
thf(fact_1130_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1131_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1132_ereal__less__eq_I6_J,axiom,
! [R: real] :
( ( ord_le1083603963089353582_ereal @ ( extended_ereal2 @ R ) @ one_on4623092294121504201_ereal )
= ( ord_less_eq_real @ R @ one_one_real ) ) ).
% ereal_less_eq(6)
thf(fact_1133_ereal__less__eq_I7_J,axiom,
! [R: real] :
( ( ord_le1083603963089353582_ereal @ one_on4623092294121504201_ereal @ ( extended_ereal2 @ R ) )
= ( ord_less_eq_real @ one_one_real @ R ) ) ).
% ereal_less_eq(7)
thf(fact_1134_ereal__plus__1_I3_J,axiom,
( ( plus_p7876563987511257093_ereal @ one_on4623092294121504201_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% ereal_plus_1(3)
thf(fact_1135_ereal__plus__1_I4_J,axiom,
( ( plus_p7876563987511257093_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ one_on4623092294121504201_ereal )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% ereal_plus_1(4)
thf(fact_1136_ereal__less_I4_J,axiom,
! [R: real] :
( ( ord_le1188267648640031866_ereal @ one_on4623092294121504201_ereal @ ( extended_ereal2 @ R ) )
= ( ord_less_real @ one_one_real @ R ) ) ).
% ereal_less(4)
thf(fact_1137_ereal__less_I3_J,axiom,
! [R: real] :
( ( ord_le1188267648640031866_ereal @ ( extended_ereal2 @ R ) @ one_on4623092294121504201_ereal )
= ( ord_less_real @ R @ one_one_real ) ) ).
% ereal_less(3)
thf(fact_1138_ereal__less__eq_I4_J,axiom,
! [R: real] :
( ( ord_le1083603963089353582_ereal @ ( extended_ereal2 @ R ) @ zero_z2744965634713055877_ereal )
= ( ord_less_eq_real @ R @ zero_zero_real ) ) ).
% ereal_less_eq(4)
thf(fact_1139_ereal__less__eq_I5_J,axiom,
! [R: real] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( extended_ereal2 @ R ) )
= ( ord_less_eq_real @ zero_zero_real @ R ) ) ).
% ereal_less_eq(5)
thf(fact_1140_ereal__plus__1_I2_J,axiom,
! [R: real] :
( ( plus_p7876563987511257093_ereal @ ( extended_ereal2 @ R ) @ one_on4623092294121504201_ereal )
= ( extended_ereal2 @ ( plus_plus_real @ R @ one_one_real ) ) ) ).
% ereal_plus_1(2)
thf(fact_1141_ereal__plus__1_I1_J,axiom,
! [R: real] :
( ( plus_p7876563987511257093_ereal @ one_on4623092294121504201_ereal @ ( extended_ereal2 @ R ) )
= ( extended_ereal2 @ ( plus_plus_real @ R @ one_one_real ) ) ) ).
% ereal_plus_1(1)
thf(fact_1142_real__of__ereal__le__0,axiom,
! [X2: extended_ereal] :
( ( ord_less_eq_real @ ( extend2982805604970551563_ereal @ X2 ) @ zero_zero_real )
= ( ( ord_le1083603963089353582_ereal @ X2 @ zero_z2744965634713055877_ereal )
| ( X2 = extend1530274965995635425_ereal ) ) ) ).
% real_of_ereal_le_0
thf(fact_1143_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1144_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1145_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1146_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1147_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1148_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1149_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1150_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1151_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1152_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1153_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1154_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1155_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1156_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1157_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1158_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1159_ereal__times_I1_J,axiom,
one_on4623092294121504201_ereal != extend1530274965995635425_ereal ).
% ereal_times(1)
thf(fact_1160_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1161_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1162_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1163_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1164_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1165_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1166_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1167_diff__less__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1168_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1169_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1170_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1171_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1172_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1173_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1174_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1175_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1176_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1177_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1178_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1179_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
& ( M3 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1180_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1181_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1182_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1183_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1184_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1185_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X: real,Y9: real] :
( ( ord_less_real @ X @ Y9 )
| ( X = Y9 ) ) ) ) ).
% less_eq_real_def
thf(fact_1186_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1187_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1188_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1189_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1190_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1191_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1192_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1193_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1194_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1195_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1196_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1197_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1198_mult__mono__nonpos__nonpos,axiom,
! [C2: real,A: real,D: real,B: real] :
( ( ord_less_eq_real @ C2 @ A )
=> ( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ D @ B )
=> ( ( ord_less_eq_real @ D @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ C2 @ D ) ) ) ) ) ) ).
% mult_mono_nonpos_nonpos
thf(fact_1199_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1200_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1201_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1202_real__of__ereal__le__1,axiom,
! [A: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ one_on4623092294121504201_ereal )
=> ( ord_less_eq_real @ ( extend2982805604970551563_ereal @ A ) @ one_one_real ) ) ).
% real_of_ereal_le_1
thf(fact_1203_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1204_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1205_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1206_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1207_sum__neq__zeroI_I2_J,axiom,
! [K: real,A: real,B: real] :
( ( ord_less_real @ K @ ( abs_abs_real @ A ) )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ B ) @ K )
=> ( ( plus_plus_real @ A @ B )
!= zero_zero_real ) ) ) ).
% sum_neq_zeroI(2)
thf(fact_1208_sum__neq__zeroI_I1_J,axiom,
! [K: real,A: real,B: real] :
( ( ord_less_eq_real @ K @ ( abs_abs_real @ A ) )
=> ( ( ord_less_real @ ( abs_abs_real @ B ) @ K )
=> ( ( plus_plus_real @ A @ B )
!= zero_zero_real ) ) ) ).
% sum_neq_zeroI(1)
thf(fact_1209_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1210_minus__real__def,axiom,
( minus_minus_real
= ( ^ [X: real,Y9: real] : ( plus_plus_real @ X @ ( uminus_uminus_real @ Y9 ) ) ) ) ).
% minus_real_def
thf(fact_1211_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1212_ereal__times_I3_J,axiom,
( one_on4623092294121504201_ereal
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% ereal_times(3)
thf(fact_1213_le__ereal__le,axiom,
! [A: extended_ereal,X2: real,Y: real] :
( ( ord_le1083603963089353582_ereal @ A @ ( extended_ereal2 @ X2 ) )
=> ( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_le1083603963089353582_ereal @ A @ ( extended_ereal2 @ Y ) ) ) ) ).
% le_ereal_le
thf(fact_1214_ereal__le__le,axiom,
! [Y: real,A: extended_ereal,X2: real] :
( ( ord_le1083603963089353582_ereal @ ( extended_ereal2 @ Y ) @ A )
=> ( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_le1083603963089353582_ereal @ ( extended_ereal2 @ X2 ) @ A ) ) ) ).
% ereal_le_le
thf(fact_1215_one__ereal__def,axiom,
( one_on4623092294121504201_ereal
= ( extended_ereal2 @ one_one_real ) ) ).
% one_ereal_def
thf(fact_1216_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1217_less__ereal__le,axiom,
! [A: extended_ereal,X2: real,Y: real] :
( ( ord_le1188267648640031866_ereal @ A @ ( extended_ereal2 @ X2 ) )
=> ( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_le1188267648640031866_ereal @ A @ ( extended_ereal2 @ Y ) ) ) ) ).
% less_ereal_le
thf(fact_1218_ereal__less__le,axiom,
! [Y: real,A: extended_ereal,X2: real] :
( ( ord_le1188267648640031866_ereal @ ( extended_ereal2 @ Y ) @ A )
=> ( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_le1188267648640031866_ereal @ ( extended_ereal2 @ X2 ) @ A ) ) ) ).
% ereal_less_le
thf(fact_1219_zero__less__one__ereal,axiom,
ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ one_on4623092294121504201_ereal ).
% zero_less_one_ereal
thf(fact_1220_one__not__le__zero__ereal,axiom,
~ ( ord_le1083603963089353582_ereal @ one_on4623092294121504201_ereal @ zero_z2744965634713055877_ereal ) ).
% one_not_le_zero_ereal
thf(fact_1221_ereal__0__less__1,axiom,
ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ one_on4623092294121504201_ereal ).
% ereal_0_less_1
thf(fact_1222_ereal__one__not__less__zero__ereal,axiom,
~ ( ord_le1188267648640031866_ereal @ one_on4623092294121504201_ereal @ zero_z2744965634713055877_ereal ) ).
% ereal_one_not_less_zero_ereal
thf(fact_1223_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1224_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1225_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1226_real__0__le__add__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X2 @ Y ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X2 ) @ Y ) ) ).
% real_0_le_add_iff
thf(fact_1227_real__add__le__0__iff,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X2 @ Y ) @ zero_zero_real )
= ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X2 ) ) ) ).
% real_add_le_0_iff
thf(fact_1228_ereal__m1__less__0,axiom,
ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ one_on4623092294121504201_ereal ) @ zero_z2744965634713055877_ereal ).
% ereal_m1_less_0
thf(fact_1229_real__of__ereal__minus_H,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( ( ( abs_ab7465543570706387889_ereal @ X2 )
= extend1530274965995635425_ereal )
= ( ( abs_ab7465543570706387889_ereal @ Y )
= extend1530274965995635425_ereal ) )
=> ( ( minus_minus_real @ ( extend2982805604970551563_ereal @ X2 ) @ ( extend2982805604970551563_ereal @ Y ) )
= ( extend2982805604970551563_ereal @ ( minus_2816186181549245109_ereal @ X2 @ Y ) ) ) ) ).
% real_of_ereal_minus'
thf(fact_1230_distrib__left__ereal__nn,axiom,
! [C2: real,X2: extended_ereal,Y: extended_ereal] :
( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ( times_7703590493115627913_ereal @ ( plus_p7876563987511257093_ereal @ X2 @ Y ) @ ( extended_ereal2 @ C2 ) )
= ( plus_p7876563987511257093_ereal @ ( times_7703590493115627913_ereal @ X2 @ ( extended_ereal2 @ C2 ) ) @ ( times_7703590493115627913_ereal @ Y @ ( extended_ereal2 @ C2 ) ) ) ) ) ).
% distrib_left_ereal_nn
thf(fact_1231_real__of__ereal__pos,axiom,
! [X2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ X2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( extend2982805604970551563_ereal @ X2 ) ) ) ).
% real_of_ereal_pos
thf(fact_1232_real__of__ereal__positive__mono,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ X2 )
=> ( ( ord_le1083603963089353582_ereal @ X2 @ Y )
=> ( ( Y != extend1530274965995635425_ereal )
=> ( ord_less_eq_real @ ( extend2982805604970551563_ereal @ X2 ) @ ( extend2982805604970551563_ereal @ Y ) ) ) ) ) ).
% real_of_ereal_positive_mono
thf(fact_1233_real__of__ereal__minus,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ( ( ( abs_ab7465543570706387889_ereal @ A )
= extend1530274965995635425_ereal )
| ( ( abs_ab7465543570706387889_ereal @ B )
= extend1530274965995635425_ereal ) )
=> ( ( extend2982805604970551563_ereal @ ( minus_2816186181549245109_ereal @ A @ B ) )
= zero_zero_real ) )
& ( ~ ( ( ( abs_ab7465543570706387889_ereal @ A )
= extend1530274965995635425_ereal )
| ( ( abs_ab7465543570706387889_ereal @ B )
= extend1530274965995635425_ereal ) )
=> ( ( extend2982805604970551563_ereal @ ( minus_2816186181549245109_ereal @ A @ B ) )
= ( minus_minus_real @ ( extend2982805604970551563_ereal @ A ) @ ( extend2982805604970551563_ereal @ B ) ) ) ) ) ).
% real_of_ereal_minus
thf(fact_1234_real__le__ereal__iff,axiom,
! [Y: extended_ereal,X2: real] :
( ( ord_less_eq_real @ ( extend2982805604970551563_ereal @ Y ) @ X2 )
= ( ( ( ( abs_ab7465543570706387889_ereal @ Y )
!= extend1530274965995635425_ereal )
=> ( ord_le1083603963089353582_ereal @ Y @ ( extended_ereal2 @ X2 ) ) )
& ( ( ( abs_ab7465543570706387889_ereal @ Y )
= extend1530274965995635425_ereal )
=> ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ) ) ).
% real_le_ereal_iff
thf(fact_1235_ereal__le__real__iff,axiom,
! [X2: real,Y: extended_ereal] :
( ( ord_less_eq_real @ X2 @ ( extend2982805604970551563_ereal @ Y ) )
= ( ( ( ( abs_ab7465543570706387889_ereal @ Y )
!= extend1530274965995635425_ereal )
=> ( ord_le1083603963089353582_ereal @ ( extended_ereal2 @ X2 ) @ Y ) )
& ( ( ( abs_ab7465543570706387889_ereal @ Y )
= extend1530274965995635425_ereal )
=> ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ) ) ).
% ereal_le_real_iff
thf(fact_1236_ereal__le__mult__one__interval,axiom,
! [Y: extended_ereal,X2: extended_ereal] :
( ( Y
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ! [Z: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ Z )
=> ( ( ord_le1188267648640031866_ereal @ Z @ one_on4623092294121504201_ereal )
=> ( ord_le1083603963089353582_ereal @ ( times_7703590493115627913_ereal @ Z @ X2 ) @ Y ) ) )
=> ( ord_le1083603963089353582_ereal @ X2 @ Y ) ) ) ).
% ereal_le_mult_one_interval
thf(fact_1237_lemma__interval__lt,axiom,
! [A: real,X2: real,B: real] :
( ( ord_less_real @ A @ X2 )
=> ( ( ord_less_real @ X2 @ B )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ! [Y11: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y11 ) ) @ D5 )
=> ( ( ord_less_real @ A @ Y11 )
& ( ord_less_real @ Y11 @ B ) ) ) ) ) ) ).
% lemma_interval_lt
thf(fact_1238_lemma__interval,axiom,
! [A: real,X2: real,B: real] :
( ( ord_less_real @ A @ X2 )
=> ( ( ord_less_real @ X2 @ B )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ! [Y11: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y11 ) ) @ D5 )
=> ( ( ord_less_eq_real @ A @ Y11 )
& ( ord_less_eq_real @ Y11 @ B ) ) ) ) ) ) ).
% lemma_interval
thf(fact_1239_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1240_zle__add1__eq__le,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_1241_zabs__less__one__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
= ( Z2 = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1242_zle__diff1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z2 @ one_one_int ) )
= ( ord_less_int @ W @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_1243_zabs__def,axiom,
( abs_abs_int
= ( ^ [I4: int] : ( if_int @ ( ord_less_int @ I4 @ zero_zero_int ) @ ( uminus_uminus_int @ I4 ) @ I4 ) ) ) ).
% zabs_def
thf(fact_1244_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1245_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( ( M = one_one_int )
& ( N = one_one_int ) )
| ( ( M
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1246_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_1247_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
= one_one_int )
=> ( ( abs_abs_int @ M )
= one_one_int ) ) ).
% abs_zmult_eq_1
thf(fact_1248_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_1249_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1250_int__induct__abs,axiom,
! [P: int > $o,J: int] :
( ! [N3: int] :
( ! [I3: int] :
( ( ord_less_int @ ( abs_abs_int @ I3 ) @ ( abs_abs_int @ N3 ) )
=> ( P @ I3 ) )
=> ( P @ N3 ) )
=> ( P @ J ) ) ).
% int_induct_abs
thf(fact_1251_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1252_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1253_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1254_one__integer_Orsp,axiom,
one_one_int = one_one_int ).
% one_integer.rsp
thf(fact_1255_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_1256_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1257_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_1258_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1259_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_1260_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1261_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1262_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1263_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1264_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1265_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1266_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1267_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1268_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_1269_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y10: nat] :
( ( P @ Y10 )
=> ( ord_less_eq_nat @ Y10 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y11: nat] :
( ( P @ Y11 )
=> ( ord_less_eq_nat @ Y11 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1270_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
% Helper facts (9)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y: int] :
( ( if_int @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y: int] :
( ( if_int @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y: real] :
( ( if_real @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y: real] :
( ( if_real @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_3_1_If_001t__Extended____Real__Oereal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Extended____Real__Oereal_T,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( if_Extended_ereal @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Extended____Real__Oereal_T,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( if_Extended_ereal @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( ( ( size_size_list_a @ xsa )
= zero_zero_nat )
=> ( ( prefix3213528784805800034_count @ ( prefix4097710381326367690Lf_e_a @ e @ zero_zero_nat @ xsa ) )
= ( groups3567983573054521703_ereal
@ ( map_a_Extended_ereal
@ ^ [X: a] : ( prefix3213528784805800034_count @ ( e @ X ) )
@ xsa ) ) ) )
& ( ( ( size_size_list_a @ xsa )
!= zero_zero_nat )
=> ( ( prefix3213528784805800034_count @ ( prefix4097710381326367690Lf_e_a @ e @ zero_zero_nat @ xsa ) )
= extend1530274965995635425_ereal ) ) ) ).
%------------------------------------------------------------------------------